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1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

matics of Paul Erdős, Vol. 2, Springer 1996. Turán's theorem is not merely one extremal result among others: it is the result that sparked off the entire line of research. Our first proof of Turán's theorem is essentially the original one; the second is a version of a proof of Zykov due to Brandt. Our version of the Erdős-Stone theorem is a slight simplification of the original. A direct proof, not using the regularity lemma, is given in L. Lovász, Combinatorial Problems and Exercises (2nd edn.), North-Holland 1993. Its most fundamental application, Corollary 7.1.3, was only found 20 years after the theorem, by Erdős and Simonovits (1966). Of our two bounds on \( \operatorname{ex}\left( {n,{K}_{r, r}}\right) \) the upper one is thought to give the correct order of magnitude. For vastly off-diagonal complete bipartite graphs this was verified by J. Kollár, L. Rónyai & T. Szabó, Norm-graphs and bipartite Turán numbers, Combinatorica 16 (1996), 399-406, who proved that \( \operatorname{ex}\left( {n,{K}_{r, s}}\right) \geq {c}_{r}{n}^{2 - \frac{1}{r}} \) when \( s > r \) !. Details about the Erdős-Sós conjecture, including an approximate solution for large \( k \), can be found in the survey by Komlós and Simonovits cited below. The case where the tree \( T \) is a path (Exercise 16) was proved by Erdős & Gallai in 1959. It was this result, together with the easy case of stars (Exercise 15) at the other extreme, that inspired the conjecture as a possible unifying result. Theorem 7.2.1 was first proved by B. Bollobás & A.G. Thomason, Proof of a conjecture of Mader, Erdős and Hajnal on topological complete subgraphs, Europ. J. Combinatorics 19 (1998), 883-887, and independently by J. Komlós & E. Szemerédi, Topological cliques in graphs II, Combinatorics, Probability and Computing \( \mathbf{5} \) (1996),79-90. For large \( G \), the latter authors show that the constant \( c \) in the theorem can be brought down to about \( \frac{1}{2} \) , which is not far from the lower bound of \( \frac{1}{8} \) given in Exercise 25. Theorem 7.2.2 was first proved in 1982 by Kostochka, and in 1984 with a better constant by Thomason. For references and more insight also in these early proofs, see A.G. Thomason, The extremal function for complete minors, J. Combin. Theory B 81 (2001), 318-338, where he determines the value of \( \alpha \) . Surprisingly, the average degree needed to force an incomplete minor \( H \) remains at \( {cr}\sqrt{\log r} \), with \( c = \alpha \sqrt{\epsilon } + o\left( 1\right) \) for almost all \( H \) with \( r \) vertices and \( {r}^{1 + \epsilon } \) edges, for every fixed \( \epsilon \in \left( {0,1}\right) \) ; see J.S. Myers &A.G. Thomason, The extremal function for noncomplete minors, Combinatorica (to appear). As Theorem 7.2.2 is best possible, there is no constant \( c \) such that all graphs of average degree at least \( {cr} \) have a \( {K}^{r} \) minor. Strengthening this assumption to \( \kappa \geq {cr} \), however, can force a \( {K}^{r} \) minor in all large enough graphs; this was proved by T. Böhme, K. Kawarabayashi, J. Maharry and B. Mohar, Linear connectivity forces large complete bipartite minors, preprint 2004. The fact that large enough girth can force minors of arbitrarily high minimum degree, and hence large complete minors, was discovered by Thomassen in 1983. The reference can be found in W. Mader, Topological subgraphs in graphs of large girth, Combinatorica 18 (1998), 405-412, from which our Lemma 7.2.3 is extracted. Our girth assumption of \( {8k} + 3 \) has been reduced to about \( {4k} \) by D. Kühn and D. Osthus, Minors in graphs of large girth, Random Struct. Alg. 22 (2003), 213-225, which is conjectured to be best possible. The original reference for Theorem 7.2.5 can be found in D. Kühn and D. Osthus, Improved bounds for topological cliques in graphs of large girth (preprint 2005), where they re-prove their theorem with \( g \leq {27} \) . See also D. Kühn &D. Osthus, Subdivisions of \( {K}_{r + 2} \) in graphs of average degree at least \( r + \varepsilon \) and large but constant girth, Combinatorics, Probability and Computing 13 (2004), 361-371. The proof of Hadwiger’s conjecture for \( r = 4 \) hinted at in Exercise 34 was given by Hadwiger himself, in the 1943 paper containing his conjecture. A counterexample to Hajós's conjecture was found as early as 1979 by Catlin. A little later, Erdős and Fajtlowicz proved that Hajós's conjecture is false for 'almost all' graphs (see Chapter 11). Proofs of Wagner's Theorem 7.3.4 (with Hadwiger’s conjecture for \( r = 5 \) as a corollary) can be found in Bollobás’s Extremal Graph Theory (see above) and in Halin's Graphentheorie (2nd ed.), Wissenschaftliche Buchgesellschaft 1989. Hadwiger’s conjecture for \( r = 6 \) was proved by N. Robertson, P.D. Seymour and R. Thomas, Hadwiger's conjecture for \( {K}_{6} \) -free graphs, Combinatorica 13 (1993),279-361. The investigation of graphs not containing a given graph as a minor, or topological minor, has a long history. It probably started with Wagner's 1935 PhD thesis, in which he sought to 'detopologize' the four colour problem by classifying the graphs without a \( {K}^{5} \) minor. His hope was to be able to show abstractly that all those graphs were 4-colourable; since the graphs without a \( {K}^{5} \) minor include the planar graphs, this would amount to a proof of the four colour conjecture involving no topology whatsoever. The result of Wagner's efforts, Theorem 7.3.4, falls tantalizingly short of this goal: although it succeeds in classifying the graphs without a \( {K}^{5} \) minor in structural terms, planarity re-emerges as one of the criteria used in the classification. From this point of view, it is instructive to compare Wagner’s \( {K}^{5} \) theorem with similar classification theorems, such as his analogue for \( {K}^{4} \) (Proposition 7.3.1), where the graphs are decomposed into parts from a finite set of irreducible graphs. See R. Diestel, Graph Decompositions, Oxford University Press 1990, for more such classification theorems. Despite its failure to resolve the four colour problem, Wagner’s \( {K}^{5} \) structure theorem had consequences for the development of graph theory like few others. To mention just two: it prompted Hadwiger to make his famous conjecture; and it inspired the notion of a tree-decomposition, which is fundamental to the work of Robertson and Seymour on minors (see Chapter 12). Wagner himself responded to Hadwiger's conjecture with a proof that, in order to force a \( {K}^{r} \) minor, it does suffice to raise the chromatic number of a graph to some value depending only on \( r \) (Exercise 19). This theorem, along with its analogue for topological minors proved independently by Dirac and by Jung, prompted the question which average degree suffices to force the desired minor. Theorem 7.3.8 is a consequence of the more fundamental result of D. Kühn and D. Osthus, Complete minors in \( {K}_{s, s} \) -free graphs, Combinatorica 25 (2005) 49-64, that every graph without a \( {K}_{s, s} \) subgraph that has average degree \( r \geq {r}_{s} \) has a \( {K}^{p} \) minor for \( p = \left\lfloor {{r}^{1 + \frac{1}{2\left( {s - 1}\right) }}/{\left( \log r\right) }^{3}}\right\rfloor \) . As in Gyárfás's conjecture, one may ask under what additional assumptions large average degree forces an induced subdivision of a given graph \( H \) . This was answered for arbitrary \( H \) by D. Kühn and D. Osthus, Induced subdivisions in \( {K}_{s, s} \) -free graphs of large average degree, Combinatorica 24 (2004) 287-304, who proved that for all \( r, s \in \mathbb{N} \) there exists \( d \in \mathbb{N} \) such that every graph \( G \nsupseteq {K}_{s, s} \) with \( d\left( G\right) \geq d \) contains a \( T{K}^{r} \) as an induced subgraph. See there also for the source of Gyárfás's conjecture and related results. The regularity lemma is proved in E. Szemerédi, Regular partitions of graphs, Colloques Internationaux CNRS 260-Problèmes Combinatoires et Théorie des Graphes, Orsay (1976), 399-401. Our rendering follows an account by Scott (personal communication). A broad survey on the regularity lemma and its applications is given by J. Komlós & M. Simonovits in (D. Miklós, V.T. Sós & T. Szőnyi, eds.) Paul Erdős is 80, Vol. 2, Proc. Colloq. Math. Soc. János Bolyai (1996); the concept of a regularity graph and Lemma 7.5.2 are taken from this paper. An adaptation of the regularity lemma for use with sparse graphs was developed independently by Kohayakawa and by Rödl; see Y. Kohayakawa, Szemerédi's regularity lemma for sparse graphs, in (F. Cucker & M. Shub, eds.) Foundations of Computational Mathematics, Selected papers of a conference held at IMPA in Rio de Janeiro, January 1997, Springer 1997. ## Infinite Graphs The study of infinite graphs is an attractive, but often neglected, part of graph theory. This chapter aims to give an introduction that starts gently, but then moves on in several directions to display both the breadth and some of the depth that this field has to offer. \( {}^{1} \) Our overall theme will be to highlight the typical kinds of phenomena that will always appear when graphs are infinite, and to show how they can lead to deep and fascinating problems. Perhaps the most typical such phenomena occur already when the graphs are 'only just' infinite, when they have only countably many vertices and perhaps only finitely many edges at each vertex. This is not surprising: after all, some of the most basic structural features of graphs, such as paths, are intrinsically countable. Problems that become really interesting only for uncountable graphs tend to be interesting for reasons that have more to do with sets than with graphs, and are studied in combinatorial set theory. This, too, is a fascinating field, but not our topic in this chapter. The problems we shall consider will all be interesting for countable graphs, and set-theor

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

ut then moves on in several directions to display both the breadth and some of the depth that this field has to offer. \( {}^{1} \) Our overall theme will be to highlight the typical kinds of phenomena that will always appear when graphs are infinite, and to show how they can lead to deep and fascinating problems. Perhaps the most typical such phenomena occur already when the graphs are 'only just' infinite, when they have only countably many vertices and perhaps only finitely many edges at each vertex. This is not surprising: after all, some of the most basic structural features of graphs, such as paths, are intrinsically countable. Problems that become really interesting only for uncountable graphs tend to be interesting for reasons that have more to do with sets than with graphs, and are studied in combinatorial set theory. This, too, is a fascinating field, but not our topic in this chapter. The problems we shall consider will all be interesting for countable graphs, and set-theoretic problems will not arise. The terminology we need is exactly the same as for finite graphs, except when we wish to describe an aspect of infinite graphs that has no finite counterpart. One important such aspect is the eventual behaviour of the infinite paths in a graph, which is captured by the notion of ends. The ends of a graph can be thought of as additional limit points at infinity to which its infinite paths converge. This convergence is described formally in terms of a natural topology placed on the graph together with its ends. In our last section we shall therefore assume familiarity with the basic concepts of point-set topology; reminders of the relevant definitions will be included as they arise. --- 1 The sections will alternate in difficulty: while Sections 8.1, 8.3 and 8.5 are easier, Sections 8.2 and 8.4 contain some more substantial proofs. --- ## 8.1 Basic notions, facts and techniques This section gives a gentle introduction to the aspects of infinity most commonly encountered in graph theory. \( {}^{2} \) After just a couple of definitions, we begin by looking at a few obvious properties of infinite sets, and how they can be employed in the context of graphs. We then illustrate how to use the three most basic common tools in infinite graph theory: Zorn's lemma, transfinite induction, and something called 'compactness'. We complete the section with the combinatorial definition of an end; topological aspects will be treated in Section 8.5. locally A graph is locally finite if all its vertices have finite degrees. An in- finite finite graph \( \left( {V, E}\right) \) of the form \[ V = \left\{ {{x}_{0},{x}_{1},{x}_{2},\ldots }\right\} \;E = \left\{ {{x}_{0}{x}_{1},{x}_{1}{x}_{2},{x}_{2}{x}_{3},\ldots }\right\} \] rays is called a ray, and a double ray is an infinite graph \( \left( {V, E}\right) \) of the form \[ V = \left\{ {\ldots ,{x}_{-1},{x}_{0},{x}_{1},\ldots }\right\} \;E = \left\{ {\ldots ,{x}_{-1}{x}_{0},{x}_{0}{x}_{1},{x}_{1}{x}_{2},\ldots }\right\} ; \] in both cases the \( {x}_{n} \) are assumed to be distinct. Thus, up to isomorphism, there is only one ray and one double ray, the latter being the unique infinite 2-regular connected graph. In the context of infinite path graphs, finite paths rays and double rays are all called paths. tail The subrays of a ray or double ray are its tails. Formally, every ray has infinitely many tails, but any two of them differ only by a finite initial segment. The union of a ray \( R \) with infinitely many disjoint finite comb paths having precisely their first vertex on \( R \) is a comb; the last vertices teeth, spine of those paths are the teeth of this comb, and \( R \) is its spine. (If such a path is trivial, which we allow, then its unique vertex lies on \( R \) and also counts as a tooth; see Figure 8.1.1.)  Fig. 8.1.1. A comb with white teeth and spine \( R = {x}_{0}{x}_{1}\ldots \) --- 2 This introductory section is deliberately kept informal, with the emphasis on ideas rather than definitions that do not belong in a graph theory book. A more formal reminder of those basic definitions about infinite sets and numbers that we shall need is given in an appendix at the end of the book. --- Let us now look at a few very basic properties of infinite sets, and see how they appear in some typical arguments about graphs. An infinite set minus a finite subset is still infinite. (1) This trivial property is eminently useful when the infinite set in question plays the role of 'supplies' that keep an iterated process going. For example, let us show that if a graph \( G \) is infinitely connected (that is, if \( G \) is \( k \) -connected for every \( k \in \mathbb{N} \) ), then \( G \) contains a subdivision of \( {K}^{{\aleph }_{0}} \), the complete graph of order \( \left| \mathbb{N}\right| \) . We embed \( {K}^{{\aleph }_{0}} \) in \( G \) (as a topological minor) in one infinite sequence \( {}^{3} \) of steps, as follows. We begin by enumerating its vertices. Then at each step we embed the next vertex in \( G \), connecting it to the images of its earlier neighbours by paths in \( G \) that avoid any other vertices used so far. The point here is that each new path has to avoid only finitely many previously used vertices, which is not a problem since deleting any finite set of vertices keeps \( G \) infinitely connected. If \( G \), too, is countable, can we then also find a \( T{K}^{{\aleph }_{0}} \) as a spanning subgraph of \( G \) ? Although embedding \( {K}^{{\aleph }_{0}} \) in \( G \) topologically as above takes infinitely many steps, it is by no means guaranteed that the \( T{K}^{{\aleph }_{0}} \) constructed uses all the vertices of \( G \) . However, it is not difficult to ensure this: since we are free to choose the image of each new vertex of \( {K}^{{\aleph }_{0}} \), we can choose this as the next unused vertex from some fixed enumeration of \( V\left( G\right) \) . In this way, every vertex of \( G \) gets chosen eventually, unless it becomes part of the \( T{K}^{{\aleph }_{0}} \) before its time, as a subdividing vertex on one of the paths. Unions of countably many countable sets are countable. (2) This fact can be applied in two ways: to show that sets that come to us as countable unions are 'small', but also to rewrite a countable set deliberately as a disjoint union of infinitely many infinite subsets. For an example of the latter type of application, let us show that an infinitely edge-connected countable graph has infinitely many edge-disjoint spanning trees. (Note that the converse implication is trivial.) The trick is to construct the trees simultaneously, in one infinite sequence of steps. We first use (2) to partition \( \mathbb{N} \) into infinitely many infinite subsets \( {N}_{i} \) \( \left( {i \in \mathbb{N}}\right) \) . Then at step \( n \) we look which \( {N}_{i} \) contains \( n \), and add a further vertex \( v \) to the \( i \) th tree \( {T}_{i} \) . As before, we choose \( v \) minimal in some fixed enumeration of \( V\left( G\right) \) among the vertices not yet in \( {T}_{i} \), and join \( v \) to \( {T}_{i} \) by a path avoiding the finitely many edges used so far. Clearly, a countable set cannot have uncountably many disjoint subsets. However, 3 We reserve the term 'infinite sequence' for sequences indexed by the set of natural numbers. (In the language of well-orderings: for sequences of order type \( \omega \) .) A countable set can have uncountably many subsets whose (3) pairwise intersections are all finite. This is a remarkable property of countable sets, and a good source of counterexamples to rash conjectures. Can you prove it without looking at Figure 8.1.4? Another common pitfall in dealing with infinite sets is to assume that the intersection of an infinite nested sequence \( {A}_{0} \supseteq {A}_{1} \supseteq \ldots \) of uncountable sets must still be uncountable. It need not be; in fact it may be empty. (Example?) There are a few basic proof techniques that are specific to infinite combinatorics. The two most common of these are the use of Zorn's lemma and transfinite induction. Rather than describing these formally, \( {}^{4} \) we illustrate their use by a simple example. Proposition 8.1.1. Every connected graph contains a spanning tree. First proof (by Zorn's lemma). Given a connected graph \( G \), consider the set of all trees \( T \subseteq G \), ordered by the subgraph relation. Since \( G \) is connected, any maximal such tree contains every vertex of \( G \), i.e. is a spanning tree of \( G \) . To prove that a maximal tree exists, we have to show that for any chain \( \mathcal{C} \) of such trees there is an upper bound: a tree \( {T}^{ * } \subseteq G \) containing every tree in \( \mathcal{C} \) as a subgraph. We claim that \( {T}^{ * } \mathrel{\text{:=}} \bigcup \mathcal{C} \) is such a tree. To show that \( {T}^{ * } \) is connected, let \( u, v \in {T}^{ * } \) be two vertices. Then in \( \mathcal{C} \) there is a tree \( {T}_{u} \) containing \( u \) and a tree \( {T}_{v} \) containing \( v \) . One of these is a subgraph of the other, say \( {T}_{u} \subseteq {T}_{v} \) . Then \( {T}_{v} \) contains a path from \( u \) to \( v \), and this path is also contained in \( {T}^{ * } \) . To show that \( {T}^{ * } \) is acyclic, suppose it contains a cycle \( C \) . Each of the edges of \( C \) lies in some tree in \( \mathcal{C} \) . These trees form a finite subchain of \( \mathcal{C} \) , which has a maximal element \( T \) . Then \( C \subseteq T \), a contradiction. Transfinite induction and recursion are very similar to finite inductive proofs and constructions, respectively. Basically, one proceeds step by step, and may at each step assume as known what was shown or constructed before. The only difference is that one may 'start again' after p

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

l{C} \) is such a tree. To show that \( {T}^{ * } \) is connected, let \( u, v \in {T}^{ * } \) be two vertices. Then in \( \mathcal{C} \) there is a tree \( {T}_{u} \) containing \( u \) and a tree \( {T}_{v} \) containing \( v \) . One of these is a subgraph of the other, say \( {T}_{u} \subseteq {T}_{v} \) . Then \( {T}_{v} \) contains a path from \( u \) to \( v \), and this path is also contained in \( {T}^{ * } \) . To show that \( {T}^{ * } \) is acyclic, suppose it contains a cycle \( C \) . Each of the edges of \( C \) lies in some tree in \( \mathcal{C} \) . These trees form a finite subchain of \( \mathcal{C} \) , which has a maximal element \( T \) . Then \( C \subseteq T \), a contradiction. Transfinite induction and recursion are very similar to finite inductive proofs and constructions, respectively. Basically, one proceeds step by step, and may at each step assume as known what was shown or constructed before. The only difference is that one may 'start again' after performing any infinite number of steps. This is formalized by the use of ordinals rather than natural numbers for counting the steps; see the appendix. Just as with finite graphs, it is usually more intuitive to construct a desired object (such as a spanning tree) step by step, rather than starting with some unknown 'maximal' object and then proving that it has the desired properties. More importantly, a step-by-step construction is 4 The appendix offers brief introductions to both, enough to enable the reader to use these tools with confidence in practice. almost always the best way to find the desired object: only later, when one understands the construction well, can one devise an inductive ordering (one whose chains have upper bounds) in which the desired objects appear as the maximal elements. Thus, although Zorn's lemma may at times provide an elegant way to wrap up a constructive proof, it cannot in general replace a good understanding of transfinite induction-just as a preference for elegant direct definitions of finite objects cannot, for a thorough understanding, replace the more pedestrian algorithmic approach. Our second proof of Proposition 8.1.1 illustrates both the constructive and the proof aspect of transfinite induction in a typical manner: we first define a subgraph \( {T}^{ * } \subseteq G \) recursively, hoping that it turns out to be a spanning tree, and then prove inductively that it is. ## Second proof (by transfinite induction). Let \( G \) be a connected graph. We define non-empty subgraphs \( {T}_{\alpha } \subseteq G \) recursively, as follows. Let \( {T}_{0} \) consist of a single vertex. Now consider an ordinal \( \alpha > 0 \) . If \( \alpha \) is a limit, we put \( {T}_{\alpha } \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{\beta < \alpha }}{T}_{\beta } \) . If \( \alpha \) is a successor, of \( \beta \) say, we check whether \( G - {T}_{\beta } = \varnothing \) . If so, we terminate the recursion and put \( {T}_{\beta } = : T \) . If not, then \( G - {T}_{\beta } \) has a vertex \( {v}_{\alpha } \) that sends an edge \( {e}_{\alpha } \) to a vertex in \( {T}_{\beta } \) . Let \( {T}_{\alpha } \) be obtained from \( {T}_{\beta } \) by adding \( {v}_{\alpha } \) and \( {e}_{\alpha } \) . This recursion terminates, since if \( {v}_{\beta + 1} \) (where \( \beta + 1 \) denotes the successor of \( \beta \) ) gets defined for all \( \beta < \gamma \) then \( \beta \mapsto {v}_{\beta + 1} \) is an injective map showing that \( \left| \gamma \right| \leq \left| G\right| \), which cannot hold for all ordinals \( \gamma \) . We now prove by induction on \( \alpha \) that every graph \( {T}_{\alpha } \) we defined is a tree. Since \( T \) is one of the \( {T}_{\alpha } \) and is, by definition, a spanning subgraph of \( G \), this will complete the proof. Let \( \alpha \) be given, and assume that every \( {T}_{\beta } \) with \( \beta < \alpha \) is a tree. If \( \alpha \) is a successor, of \( \beta \) say, then \( {T}_{\alpha } \) is clearly connected and acyclic, because \( {T}_{\beta } \) is. Suppose now that \( \alpha \) is a limit. To show that \( {T}_{\alpha } \) is connected, let \( u, v \) be any two of its vertices. Since \( {T}_{\alpha } = \mathop{\bigcup }\limits_{{\beta < \alpha }}{T}_{\beta } \), there exist \( \beta \left( u\right) ,\beta \left( v\right) < \alpha \) such that \( u \in {T}_{\beta \left( u\right) } \) and \( v \in {T}_{\beta \left( v\right) } \), say with \( \beta \left( u\right) \leq \beta \left( v\right) \) . Then \( {T}_{\beta \left( v\right) } \) contains a \( u - v \) path, which is also contained in \( {T}_{\alpha } \) . Now suppose that \( {T}_{\alpha } \) contains a cycle \( C \) . For each of its vertices \( v \) there is an ordinal \( \beta \left( v\right) < \alpha \) with \( v \in {T}_{\beta \left( v\right) } \) ; let \( \beta \) be the largest among these. Then \( C \subseteq {T}_{\beta } \), contradicting our assumption that \( {T}_{\beta } \) is a tree. Why did these proofs work so smoothly? The reason is that the forbidden or required substructures, cycles and connecting paths, were finite and therefore could not arise or vanish unexpectedly at limit steps. This has helped to keep our two model proofs simple, but it is not typical. If we want to construct a rayless graph, for example, the edges of different rayless graphs \( {G}_{\beta } \) might combine to form a ray in \( {G}_{\alpha } = \mathop{\bigcup }\limits_{{\beta < \alpha }}{G}_{\beta } \) when \( \alpha \) is a limit. And indeed, here lies the challenge in most transfinite constructions: to make the right choices at successor steps to ensure that the structure will also be as desired at limits. Our third basic proof technique, somewhat mysteriously referred to --- compactness proofs --- as compactness (see below for why), offers a formalized way to make the right choices in certain standard cases. These are cases where, unlike in the above examples, a wrong choice may necessarily lead to a dead end after another finite number of steps, even though nothing unexpected happens at limits. For example, let \( G \) be a graph whose finite subgraphs are all \( k \) - colourable. It is natural then to try to construct a \( k \) -colouring of \( G \) as a limit of \( k \) -colourings of its finite subgraphs. Now each finite subgraph will have several \( k \) -colourings; will it matter which we choose? Clearly, it will. When \( {G}^{\prime } \subseteq {G}^{\prime \prime } \) are two finite subgraphs and \( u, v \) are vertices of \( {G}^{\prime } \) that receive the same colour in every \( k \) -colouring of \( {G}^{\prime \prime } \) (and hence also in any \( k \) -colouring of \( G \) ), we must not give them different colours in the colouring we choose for \( {G}^{\prime } \), even if such a colouring exists. However if we do manage, somehow, to colour the finite subgraphs of \( G \) compatibly, we shall automatically have a colouring of all of \( G \) . For countable graphs, compactness proofs are formalized by the following lemma: ## Lemma 8.1.2. (König's Infinity Lemma) Let \( {V}_{0},{V}_{1},\ldots \) be an infinite sequence of disjoint non-empty finite sets, and let \( G \) be a graph on their union. Assume that every vertex \( v \) in a set \( {V}_{n} \) with \( n \geq 1 \) has a neighbour \( f\left( v\right) \) in \( {V}_{n - 1} \) . Then \( G \) contains a ray \( {v}_{0}{v}_{1}\ldots \) with \( {v}_{n} \in {V}_{n} \) for all \( n \) .  Fig. 8.1.2. König's infinity lemma Proof. Let \( \mathcal{P} \) be the set of all finite paths of the form \( {vf}\left( v\right) f\left( {f\left( v\right) }\right) \ldots \) ending in \( {V}_{0} \) . Since \( {V}_{0} \) is finite but \( \mathcal{P} \) is infinite, infinitely many of the paths in \( \mathcal{P} \) end at the same vertex \( {v}_{0} \in {V}_{0} \) . Of these paths, infinitely many also agree on their penultimate vertex \( {v}_{1} \in {V}_{1} \), because \( {V}_{1} \) is finite. Of those paths, infinitely many agree even on their vertex \( {v}_{2} \) in \( {V}_{2} \) - and so on. Although the set of paths considered decreases from step to step, it is still infinite after any finite number of steps, so \( {v}_{n} \) gets defined for every \( n \in \mathbb{N} \) . By definition, each vertex \( {v}_{n} \) is adjacent to \( {v}_{n - 1} \) on one of those paths, so \( {v}_{0}{v}_{1}\ldots \) is indeed a ray. The following 'compactness theorem', the first of its kind in graph theory, answers our question about colourings: Theorem 8.1.3. (de Bruijn & Erdős, 1951) Let \( G = \left( {V, E}\right) \) be a graph and \( k \in \mathbb{N} \) . If every finite subgraph of \( G \) has chromatic number at most \( k \), then so does \( G \) . First proof (for \( G \) countable, by the infinity lemma). Let \( {v}_{0},{v}_{1},\ldots \) be an enumeration of \( V \) and put \( {G}_{n} \mathrel{\text{:=}} G\left\lbrack {{v}_{0},\ldots ,{v}_{n}}\right\rbrack \) . Write \( {V}_{n} \) for the set of all \( k \) -colourings of \( {G}_{n} \) with colours in \( \{ 1,\ldots, k\} \) . Define a graph on \( \mathop{\bigcup }\limits_{{n \in \mathbb{N}}}{V}_{n} \) by inserting all edges \( c{c}^{\prime } \) such that \( c \in {V}_{n} \) and \( {c}^{\prime } \in {V}_{n - 1} \) is the restriction of \( c \) to \( \left\{ {{v}_{0},\ldots ,{v}_{n - 1}}\right\} \) . Let \( {c}_{0}{c}_{1}\ldots \) be a ray in this graph with \( {c}_{n} \in {V}_{n} \) for all \( n \) . Then \( c \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{n \in \mathbb{N}}}{c}_{n} \) is a colouring of \( G \) with colours in \( \{ 1,\ldots, k\} \) . Our second proof of Theorem 8.1.3 appeals directly to compactness as defined in topology. Recall that a topological space is compact if its closed sets have the 'finite intersection property', which means that the overall intersection \( \bigcap \mathca

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d put \( {G}_{n} \mathrel{\text{:=}} G\left\lbrack {{v}_{0},\ldots ,{v}_{n}}\right\rbrack \) . Write \( {V}_{n} \) for the set of all \( k \) -colourings of \( {G}_{n} \) with colours in \( \{ 1,\ldots, k\} \) . Define a graph on \( \mathop{\bigcup }\limits_{{n \in \mathbb{N}}}{V}_{n} \) by inserting all edges \( c{c}^{\prime } \) such that \( c \in {V}_{n} \) and \( {c}^{\prime } \in {V}_{n - 1} \) is the restriction of \( c \) to \( \left\{ {{v}_{0},\ldots ,{v}_{n - 1}}\right\} \) . Let \( {c}_{0}{c}_{1}\ldots \) be a ray in this graph with \( {c}_{n} \in {V}_{n} \) for all \( n \) . Then \( c \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{n \in \mathbb{N}}}{c}_{n} \) is a colouring of \( G \) with colours in \( \{ 1,\ldots, k\} \) . Our second proof of Theorem 8.1.3 appeals directly to compactness as defined in topology. Recall that a topological space is compact if its closed sets have the 'finite intersection property', which means that the overall intersection \( \bigcap \mathcal{A} \) of a set \( \mathcal{A} \) of closed sets is non-empty whenever every finite subset of \( \mathcal{A} \) has a non-empty intersection. By Tychonoff’s theorem of general topology, any product of compact spaces is compact in the usual product topology. Second proof (for \( G \) arbitrary, by Tychonoff’s theorem). Consider the product space \[ X \mathrel{\text{:=}} \mathop{\prod }\limits_{V}\{ 1,\ldots, k\} = \{ 1,\ldots, k{\} }^{V} \] of \( \left| V\right| \) copies of the finite set \( \{ 1,\ldots, k\} \) endowed with the discrete topology. By Tychonoff's theorem, this is a compact space. Its basic open sets have the form \[ {O}_{h} \mathrel{\text{:=}} \left\{ {f \in X : {\left. f\right| }_{U} = h}\right\} \] where \( h \) is some map from a finite set \( U \subseteq V \) to \( \{ 1,\ldots, k\} \) . For every finite set \( U \subseteq V \), let \( {A}_{U} \) be the set of all \( f \in X \) whose restriction to \( U \) is a \( k \) -colouring of \( G\left\lbrack U\right\rbrack \) . These sets \( {A}_{U} \) are closed (as well as open-why?), and for any finite set \( \mathcal{U} \) of finite subsets of \( V \) we have \( \mathop{\bigcap }\limits_{{U \in \mathcal{U}}}{A}_{U} \neq \varnothing \), because \( G\left\lbrack {\bigcup \mathcal{U}}\right\rbrack \) has a \( k \) -colouring. By the finite intersection property of the sets \( {A}_{U} \), their overall intersection is nonempty, and every element of this intersection is a \( k \) -colouring of \( G \) . Although our two compactness proofs look formally different, it is instructive to compare them in detail, checking how the requirements in one are reflected in the other (cf. Exercise 10). As the reader may expect, the standard use for compactness proofs is to transfer theorems from finite to infinite graphs, or conversely. This is not always quite as straightforward as above; often, the statement has to be modified a little to make it susceptible to a compactness argument. As an example - see Exercises 12-17 for more - let us prove the locally finite version of the following famous conjecture. Call a bipartition of the vertex set of a graph unfriendly if every vertex has at least as many neighbours in the other class as in its own. Clearly, every finite graph has an unfriendly partition: just take any partition that maximizes the number of edges between the partition classes. At the other extreme, it can be shown by set-theoretic methods that uncountable graphs need not have such partitions. Thus, intriguingly, it is the countable case that has remained unsolved: Unfriendly Partition Conjecture. Every countable graph admits an unfriendly partition of its vertex set. Proof for locally finite graphs. Let \( G = \left( {V, E}\right) \) be an infinite but locally finite graph, and enumerate its vertices as \( {v}_{0},{v}_{1},\ldots \) For every \( n \in \mathbb{N} \) , let \( {\mathcal{V}}_{n} \) be the set of partitions of \( {V}_{n} \mathrel{\text{:=}} \left\{ {{v}_{0},\ldots ,{v}_{n}}\right\} \) into two sets \( {U}_{n} \) and \( {W}_{n} \) such that every vertex \( v \in {V}_{n} \) with \( {N}_{G}\left( v\right) \subseteq {V}_{n} \) has at least as many neighbours in the other class as in its own. Since the conjecture holds for finite graphs, the sets \( {\mathcal{V}}_{n} \) are non-empty. For all \( n \geq 1 \), every \( \left( {{U}_{n},{W}_{n}}\right) \in {\mathcal{V}}_{n} \) induces a partition \( \left( {{U}_{n - 1},{W}_{n - 1}}\right) \) of \( {V}_{n - 1} \), which lies in \( {\mathcal{V}}_{n - 1} \) . By the infinity lemma, there is an infinite sequence of partitions \( \left( {{U}_{n},{W}_{n}}\right) \in {\mathcal{V}}_{n} \), one for every \( n \in \mathbb{N} \), such that each is induced by the next. Then \( \left( {\mathop{\bigcup }\limits_{{n \in \mathbb{N}}}{U}_{n},\mathop{\bigcup }\limits_{{n \in \mathbb{N}}}{W}_{n}}\right) \) is an unfriendly partition of \( G \) . The trick that made this proof possible was to require, for the partitions of \( {V}_{n} \), correct positions only of vertices that send no edge out of \( {V}_{n} \) : this weakening is necessary to ensure that partitions from \( {\mathcal{V}}_{n} \) induce partitions in \( {\mathcal{V}}_{n - 1} \) ; but since, by local finiteness, every vertex has this property eventually (for large enough \( n \) ), the weaker assumption suffices to ensure that the limit partition is unfriendly. Let us complete this section with an introduction to the one important concept of infinite graph theory that has no finite counterpart, the notion of an end. An \( {en}{d}^{5} \) of a graph \( G \) is an equivalence class of rays in \( G \), where two rays are considered equivalent if, for every finite set \( S \subseteq V\left( G\right) \), both have a tail in the same component of \( G - S \) . This is indeed an equivalence relation: note that, since \( S \) is finite, there is exactly one such component for each ray. If two rays are equivalent-and only then - they can be linked by infinitely many disjoint paths: just 5 Not to be confused with the ends, or endvertices, of an edge. In the context of infinite graphs, we use the term 'endvertices' to avoid confusion. choose these inductively, taking as \( S \) the union of the vertex sets of the first finitely many paths to find the next. The set of ends of \( G \) is denoted by \( \Omega \left( G\right) \), and we write \( G = \left( {V, E,\Omega }\right) \) to express that \( G \) has vertex, edge \( \Omega \left( G\right) \) and end sets \( V, E,\Omega \) . For example, let us determine the ends of the 2-way infinite ladder shown in Figure 8.1.3. Every ray in this graph contains vertices arbitrarily far to the left or vertices arbitrarily far to the right, but not both. These two types of rays are clearly equivalence classes, so the ladder has exactly two ends. (In Figure 8.1.3 these are shown as two isolated dots - one on the left, the other on the right.)  Fig. 8.1.3. The 2-way ladder has two ends The ends of a tree are particularly simple: two rays in a tree are equivalent if and only if they share a tail, and for every fixed vertex \( v \) each end contains exactly one ray starting at \( v \) . Even a locally finite tree can have uncountably many ends. The prototype example (see Exercise 21) is the binary tree \( {T}_{2} \), the rooted tree in which every vertex has exactly binary tree \( {T}_{2} \) two upper neighbours. Often, the vertex set of \( {T}_{2} \) is taken to be the set of finite \( 0 - 1 \) sequences (with the empty sequence as the root), as indicated in Figure 8.1.4. The ends of \( {T}_{2} \) then correspond bijectively to its rays starting at \( \varnothing \), and hence to the infinite \( 0 - 1 \) sequences.  Fig. 8.1.4. The binary tree \( {T}_{2} \) has continuum many ends, one for every infinite \( 0 - 1 \) sequence These examples suggest that the ends of a graph can be thought of as 'points at infinity' to which its rays converge. We shall formalize this in Section 8.5, where we define a natural topology on a graph and its ends in which rays will indeed converge to their respective ends. The maximum number of disjoint rays in an end is the (combina- --- end degrees --- torial) vertex-degree of that end, the maximum number of edge-disjoint rays in it is its (combinatorial) edge-degree. These maxima are indeed attained: if an end contains a set of \( k \) (edge-) disjoint rays for every integer \( k \), it also contains an infinite set of (edge-) disjoint rays (Exercise 33). Thus, every end has a vertex-degree and an edge-degree in \( \mathbb{N} \cup \{ \infty \} \) . ## 8.2 Paths, trees, and ends There are two fundamentally different aspects to the infinity of an infinite connected graph: one of 'length', expressed in the presence of rays, and one of 'width', expressed locally by infinite degrees. The infinity lemma tells us that at least one of these must occur: Proposition 8.2.1. Every infinite connected graph has a vertex of infinite degree or contains a ray. \( \left( {8.1.2}\right) \) Proof. Let \( G \) be an infinite connected graph with all degrees finite. Let \( {v}_{0} \) be a vertex, and for every \( n \in \mathbb{N} \) let \( {V}_{n} \) be the set of vertices at distance \( n \) from \( {v}_{0} \) . Induction on \( n \) shows that the sets \( {V}_{n} \) are finite, and hence that \( {V}_{n + 1} \neq \varnothing \) (because \( G \) is infinite and connected). Furthermore, the neighbour of a vertex \( v \in {V}_{n + 1} \) on any shortest \( v - {v}_{0} \) path lies in \( {V}_{n} \) . By Lemma 8.1.2, \( G \) contains a ray. Often it is useful to have more detailed information on how this ray or vertex of infinite degree lies in \( G \) . The following lemma enables us to find it 'close to' any give

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e of rays, and one of 'width', expressed locally by infinite degrees. The infinity lemma tells us that at least one of these must occur: Proposition 8.2.1. Every infinite connected graph has a vertex of infinite degree or contains a ray. \( \left( {8.1.2}\right) \) Proof. Let \( G \) be an infinite connected graph with all degrees finite. Let \( {v}_{0} \) be a vertex, and for every \( n \in \mathbb{N} \) let \( {V}_{n} \) be the set of vertices at distance \( n \) from \( {v}_{0} \) . Induction on \( n \) shows that the sets \( {V}_{n} \) are finite, and hence that \( {V}_{n + 1} \neq \varnothing \) (because \( G \) is infinite and connected). Furthermore, the neighbour of a vertex \( v \in {V}_{n + 1} \) on any shortest \( v - {v}_{0} \) path lies in \( {V}_{n} \) . By Lemma 8.1.2, \( G \) contains a ray. Often it is useful to have more detailed information on how this ray or vertex of infinite degree lies in \( G \) . The following lemma enables us to find it 'close to' any given infinite set of vertices. \( \left\lbrack {8.5.5}\right\rbrack \) Lemma 8.2.2. (Star-Comb Lemma) Let \( U \) be an infinite set of vertices in a connected graph \( G \) . Then \( G \) contains either a comb with all teeth in \( U \) or a subdivision of an infinite star with all leaves in \( U \) . Proof. As \( G \) is connected, it contains a path between two vertices in \( U \) . This path is a tree \( T \subseteq G \) every edge of which lies on a path in \( T \) between two vertices in \( U \) . By Zorn’s lemma there is a maximal such tree \( {T}^{ * } \) . Since \( U \) is infinite and \( G \) is connected, \( {T}^{ * } \) is infinite. If \( {T}^{ * } \) has a vertex of infinite degree, it contains the desired subdivided star. Suppose now that \( {T}^{ * } \) is locally finite. Then \( {T}^{ * } \) contains a ray \( R \) (Proposition 8.2.1). Let us construct a sequence \( {P}_{1},{P}_{2},\ldots \) of disjoint \( R - U \) paths in \( {T}^{ * } \) . Having chosen \( {P}_{i} \) for every \( i < n \) for some \( n \), pick \( v \in R \) so that \( {vR} \) meets none of those paths \( {P}_{i} \) . The first edge of \( {vR} \) lies on a path \( P \) in \( {T}^{ * } \) between two vertices in \( U \) ; let us think of \( P \) as traversing this edge in the same direction as \( R \) . Let \( w \) be the last vertex of \( {vP} \) on \( {vR} \) . Then \( {P}_{n} \mathrel{\text{:=}} {wP} \) contains an \( R - U \) path, and \( {P}_{n} \cap {P}_{i} = \varnothing \) for all \( i < n \) because \( {P}_{i} \cup {Rw} \cup {P}_{n} \) contains no cycle. We shall often apply Lemma 8.2.2 in locally finite graphs, in which case it always yields a comb. Recall that a rooted tree \( T \subseteq G \) is normal in \( G \) if the endvertices of every \( T \) -path in \( G \) are comparable in the tree-order of \( T \) . If \( T \) is a spanning tree, the only \( T \) - paths are edges of \( G \) that are not edges of \( T \) . Normal spanning trees are perhaps the single most important structural tool in infinite graph theory. As in finite graphs, they exhibit the separation properties of the graph they span. \( {}^{6} \) Moreover, their normal rays, those that start at the root, reflect its end structure: normal ray Lemma 8.2.3. If \( T \) is a normal spanning tree of \( G \), then every end of \( \left\lbrack {8.5.7}\right\rbrack \) \( G \) contains exactly one normal ray of \( T \) . Proof. Let \( \omega \in \Omega \left( G\right) \) be given. Apply the star-comb lemma in \( T \) with \( \left( {1.5.5}\right) \) \( U \) the vertex set of a ray \( R \in \omega \) . If the lemma gives a subdivided star with leaves in \( U \) and centre \( z \), say, then the finite down-closure \( \lceil z\rceil \) of \( z \) in \( T \) separates infinitely many vertices \( u > z \) of \( U \) pairwise in \( G \) (Lemma 1.5.5). This contradicts our choice of \( U \) . So \( T \) contains a comb with teeth on \( R \) . Let \( {R}^{\prime } \subseteq T \) be its spine. Since every ray in \( T \) has an increasing tail (Exercise 4), we may assume that \( {R}^{\prime } \) is a normal ray. Since \( {R}^{\prime } \) is equivalent to \( R \), it lies in \( \omega \) . Conversely, distinct normal rays of \( T \) are separated in \( G \) by the (finite) down-closure of their greatest common vertex (Lemma 1.5.5), so they cannot belong to the same end of \( G \) . Not all connected graphs have a normal spanning tree; complete uncountable graphs, for example, have none. (Why not?) The quest to characterize the graphs that have a normal spanning tree is not entirely over, and it has held some surprises. \( {}^{7} \) One of the most useful sufficient conditions is that the graph contains no \( T{K}^{{\aleph }_{0}} \) ; see Theorem 12.4.13. For our purposes, the following result suffices: Theorem 8.2.4. (Jung 1967) \( \left\lbrack {8.5.9}\right\rbrack \) Every countable connected graph has a normal spanning tree. Proof. The proof follows that of Proposition 1.5.6; we only sketch the (1.5.6) differences. Starting with a single vertex, we construct an infinite se- 6 Lemma 1.5.5 continues to hold for infinite graphs, with the same proof. 7 One of these is Theorem 8.5.2; for more see the notes. quence \( {T}_{0} \subseteq {T}_{1} \subseteq \ldots \) of finite normal trees in \( G \), all with the same root, whose union \( T \) will be a normal spanning tree. To ensure that \( T \) spans \( G \), we fix an enumeration \( {v}_{0},{v}_{1},\ldots \) of \( V\left( G\right) \) and see to it that \( {T}_{n} \) contains \( {v}_{n} \) . It is clear that \( T \) will be a tree (since any cycle in \( T \) would lie in some \( {T}_{n} \), and every two vertices of \( T \) lie in a common \( {T}_{n} \) and can be linked there), and clearly the tree order of \( T \) induces that of the \( {T}_{n} \) . Finally, \( T \) will be normal, because the endvertices of any edge of \( G \) that is not an edge of \( T \) lie in some \( {T}_{n} \) : since that \( {T}_{n} \) is normal, they must be comparable there, and hence in \( T \) . It remains to specify how to construct \( {T}_{n + 1} \) from \( {T}_{n} \) . If \( {v}_{n + 1} \in {T}_{n} \) , put \( {T}_{n + 1} \mathrel{\text{:=}} {T}_{n} \) . If not, let \( C \) be the component of \( G - {T}_{n} \) containing \( {v}_{n + 1} \) . Let \( x \) be the greatest element of the chain \( N\left( C\right) \) in \( {T}_{n} \), and let \( {T}_{n + 1} \) be the union of \( {T}_{n} \) and an \( x - {v}_{n + 1} \) path \( P \) with \( P \subseteq C \) . Then the neighbourhood in \( {T}_{n + 1} \) of any new component \( {C}^{\prime } \subseteq C \) of \( G - {T}_{n + 1} \) is a chain in \( {T}_{n + 1} \), so \( {T}_{n + 1} \) is again normal. One of the most basic problems in an infinite setting that has no finite equivalent is whether or not 'arbitrarily many', in some context, implies ’infinitely many’. Suppose we can find \( k \) disjoint rays in some given graph \( G \), for every \( k \in \mathbb{N} \) ; does \( G \) also contain an infinite set of disjoint rays? The answer to the corresponding question for finite paths (of any fixed length) is clearly ’yes’, since a finite path \( P \) can never get in the way of more than \( \left| P\right| \) disjoint other paths. A badly chosen ray, however, can meet infinitely many other rays, preventing them from being selected for the same disjoint set. Rather than collecting our disjoint rays greedily, we therefore have to construct them carefully and all simultaneously. The proof of the following theorem is a nice example of a construction in an infinite sequence of steps, where the final object emerges only at the limit step. Each of the steps in the sequence will involve a nontrivial application of Menger's theorem (3.3.1). ## Theorem 8.2.5. (Halin 1965) (i) If an infinite graph \( G \) contains \( k \) disjoint rays for every \( k \in \mathbb{N} \) , then \( G \) contains infinitely many disjoint rays. (ii) If an infinite graph \( G \) contains \( k \) edge-disjoint rays for every \( k \in \mathbb{N} \) , then \( G \) contains infinitely many edge-disjoint rays. (3.3.1) Proof. (i) We construct our infinite system of disjoint rays inductively in \( \omega \) steps. After step \( n \), we shall have found \( n \) disjoint rays \( {R}_{1}^{n},\ldots ,{R}_{n}^{n} \) and chosen initial segments \( {R}_{i}^{n}{x}_{i}^{n} \) of these rays. In step \( n + 1 \) we choose the rays \( {R}_{1}^{n + 1},\ldots ,{R}_{n + 1}^{n + 1} \) so as to extend these initial segments, i.e. so that \( {R}_{i}^{n}{x}_{i}^{n} \) is a proper initial segment of \( {R}_{i}^{n + 1}{x}_{i}^{n + 1} \), for \( i = 1,\ldots, n \) . Then, clearly, the graphs \( {R}_{i}^{ * } \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{n \in \mathbb{N}}}{R}_{i}^{n}{x}_{i}^{n} \) will form an infinite family \( {\left( {R}_{i}^{ * }\right) }_{i \in \mathbb{N}} \) of disjoint rays in \( G \) . For \( n = 0 \) the empty set of rays is as required. So let us assume that \( {R}_{1}^{n},\ldots ,{R}_{n}^{n} \) have been chosen, and describe step \( n + 1 \) . For simplicity, let us abbreviate \( {R}_{i}^{n} = : {R}_{i} \) and \( {x}_{i}^{n} = : {x}_{i} \) . Let \( \mathcal{R} \) be any set of \( {R}_{i},{x}_{i} \) \( \left| {{R}_{1}{x}_{1} \cup \ldots \cup {R}_{n}{x}_{n}}\right| + {n}^{2} + 1 \) disjoint rays (which exists by assumption), and immediately delete those rays from \( \mathcal{R} \) that meet any of the paths \( {R}_{1}{x}_{1},\ldots ,{R}_{n}{x}_{n} \) ; then \( \mathcal{R} \) still contains at least \( {n}^{2} + 1 \) rays. We begin by repeating the following step as often as possible. If there exists an \( i \in \{ 1,\ldots, n\} \) such that \( {R}_{i}^{n + 1} \) has not yet been defined and \( {\mathring{x}}_{i}{R}_{i} \) meets at most \( n \) of the rays currently in \( \mathcal{R} \), we delete those rays from \( \mathcal{R} \), put \( {R}_{i}^{n + 1} \mathrel{\text{:=}} {R}_{i} \), and choose as \( {x}_{i}^{n + 1} \) t

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et us assume that \( {R}_{1}^{n},\ldots ,{R}_{n}^{n} \) have been chosen, and describe step \( n + 1 \) . For simplicity, let us abbreviate \( {R}_{i}^{n} = : {R}_{i} \) and \( {x}_{i}^{n} = : {x}_{i} \) . Let \( \mathcal{R} \) be any set of \( {R}_{i},{x}_{i} \) \( \left| {{R}_{1}{x}_{1} \cup \ldots \cup {R}_{n}{x}_{n}}\right| + {n}^{2} + 1 \) disjoint rays (which exists by assumption), and immediately delete those rays from \( \mathcal{R} \) that meet any of the paths \( {R}_{1}{x}_{1},\ldots ,{R}_{n}{x}_{n} \) ; then \( \mathcal{R} \) still contains at least \( {n}^{2} + 1 \) rays. We begin by repeating the following step as often as possible. If there exists an \( i \in \{ 1,\ldots, n\} \) such that \( {R}_{i}^{n + 1} \) has not yet been defined and \( {\mathring{x}}_{i}{R}_{i} \) meets at most \( n \) of the rays currently in \( \mathcal{R} \), we delete those rays from \( \mathcal{R} \), put \( {R}_{i}^{n + 1} \mathrel{\text{:=}} {R}_{i} \), and choose as \( {x}_{i}^{n + 1} \) the successor of \( {x}_{i} \) on \( {R}_{i} \) . Having performed this step as often as possible, we let \( I \) denote the set of those \( i \in \{ 1,\ldots, n\} \) for which \( {R}_{i}^{n + 1} \) is still undefined, and put \( \left| I\right| = : m \) . Then \( \mathcal{R} \) still contains at least \( {n}^{2} + 1 - \left( {n - m}\right) n \geq {m}^{2} + 1 \) rays. Every \( {R}_{i} \) with \( i \in I \) meets more than \( n \geq m \) of the rays in \( \mathcal{R} \) ; let \( {z}_{i} \) be its first vertex on the \( m \) th ray it meets. Then \( Z \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{i \in I}}{x}_{i}{R}_{i}{z}_{i} \) meets at most \( {m}^{2} \) of the rays in \( \mathcal{R} \) ; we delete all the other rays from \( \mathcal{R} \) , choosing one of them as \( {R}_{n + 1}^{n + 1} \) (with \( {x}_{n + 1}^{n + 1} \) arbitrary). On each remaining ray \( R \in \mathcal{R} \) we now pick a vertex \( y = y\left( R\right) \) after its last vertex in \( Z \), and put \( Y \mathrel{\text{:=}} \{ y\left( R\right) \mid R \in \mathcal{R}\} \) . Let \( H \) be the union of \( Z \) and all the paths \( {Ry}\left( {R \in \mathcal{R}}\right) \) . Then \( X \mathrel{\text{:=}} \left\{ {{x}_{i} \mid i \in I}\right\} \) cannot be separated from \( Y \) in \( H \) by fewer than \( m \) vertices, because these would miss both one of the \( m \) rays \( {R}_{i} \) with \( i \in I \) and one of the \( m \) rays in \( \mathcal{R} \) that meet \( {x}_{i}{R}_{i}{z}_{i} \) for this \( i \) . So by Menger’s theorem (3.3.1) there are \( m \) disjoint \( X - Y \) paths \( {P}_{i} = {x}_{i}\ldots {y}_{i}\left( {i \in I}\right) \) in \( H \) . For each \( i \in I \) let \( {R}_{i}^{\prime } \) denote the ray from \( \mathcal{R} \) that contains \( {y}_{i} \), choose as \( {R}_{i}^{n + 1} \) the ray \( {R}_{i}{x}_{i}{P}_{i}{y}_{i}{R}_{i}^{\prime } \), and put \( {x}_{i}^{n + 1} \mathrel{\text{:=}} {y}_{i} \) . (ii) is analogous. Does Theorem 8.2.5 generalize to other graphs than rays? Let us call a graph \( H \) ubiquitous with respect to a relation \( \leq \) between graphs (such as the subgraph relation \( \subseteq \), or the minor relation \( \preccurlyeq \) ) if \( {nH} \leq G \) for all \( n \in \mathbb{N} \) implies \( {\aleph }_{0}H \leq G \), where \( {nH} \) denotes the disjoint union of \( n \) copies of \( H \) . Ubiquity appears to be closely related to questions of well-quasi-ordering as discussed in Chapter 12. Non-ubiquitous graphs exist for all the standard graph orderings; see Exercise 36 for an example of a locally finite graph that is not ubiquitous under the subgraph relation. ## Ubiquity conjecture. (Andreae 2002) Every locally finite connected graph is ubiquitous with respect to the minor relation. Just as in Theorem 8.2.5 one can show that an end contains infinitely many disjoint rays as soon as the number of disjoint rays in it is not finitely bounded, and similarly for edge-disjoint rays (Exercise 33). Hence, the maxima in our earlier definitions of the vertex- and edge-degrees of an end exist as claimed. Ends of infinite vertex-degree are --- thick/thin --- called thick; ends of finite vertex-degree are thin. The \( \mathbb{N} \times \mathbb{N} \) grid, for example, the graph on \( {\mathbb{N}}^{2} \) in which two vertices grid \( \left( {n, m}\right) \) and \( \left( {{n}^{\prime },{m}^{\prime }}\right) \) are adjacent if and only if \( \left| {n - {n}^{\prime }}\right| + \left| {m - {m}^{\prime }}\right| = 1 \) , has only one end, which is thick. In fact, the \( \mathbb{N} \times \mathbb{N} \) grid is a kind of prototype for thick ends: every graph with a thick end contains it as a minor. This is another classical result of Halin, which we prove in the remainder of this section. For technical reasons, we shall prove Halin's theorem for hexagonal rather than square grids. These may seem a little unwieldy at first, but have the advantage that they can be found as topological rather than ordinary minors (Proposition 1.7.2), which makes them much easier to handle. We shall define the hexagonal grid \( {H}^{\infty } \) so that it is a subgraph of the \( \mathbb{N} \times \mathbb{N} \) grid, and it will be easy to see that, conversely, the \( \mathbb{N} \times \mathbb{N} \) grid is a minor of \( {H}^{\infty } \) (cf. Ex. 47, Ch. 12.) \( {H}^{\infty } \) To define our standard copy of the hexagonal quarter grid \( {H}^{\infty } \), we delete from the \( \mathbb{N} \times \mathbb{N} \) grid \( H \) the vertex \( \left( {0,0}\right) \), the vertices \( \left( {n, m}\right) \) with \( n > m \), and all edges \( \left( {n, m}\right) \left( {n + 1, m}\right) \) such that \( n \) and \( m \) have equal parity (Fig. 8.2.1). Thus, \( {H}^{\infty } \) consists of the vertical rays \( {U}_{n} \) \[ {U}_{0} \mathrel{\text{:=}} H\left\lbrack {\{ \left( {0, m}\right) \mid 1 \leq m\} }\right\rbrack \] \[ {U}_{n} \mathrel{\text{:=}} H\left\lbrack {\{ \left( {n, m}\right) \mid n \leq m\} }\right\rbrack \;\left( {n \geq 1}\right) \] and between these a set of horizontal edges, \[ E \mathrel{\text{:=}} \{ \left( {n, m}\right) \left( {n + 1, m}\right) \mid n ≢ m\left( {\;\operatorname{mod}\;2}\right) \} . \] \( {e}_{1},{e}_{2},\ldots \) To enumerate these edges, as \( {e}_{1},{e}_{2},\ldots \) say, we order them colexicograph-ically: the edge \( \left( {n, m}\right) \left( {n + 1, m}\right) \) precedes the edge \( \left( {{n}^{\prime },{m}^{\prime }}\right) \left( {{n}^{\prime } + 1,{m}^{\prime }}\right) \) if \( m < {m}^{\prime } \), or if \( m = {m}^{\prime } \) and \( n < {n}^{\prime } \) (Fig. 8.2.1). Theorem 8.2.6. (Halin 1965) Whenever a graph contains a thick end, it has a \( T{H}^{\infty } \) subgraph whose rays belong to that end. (8.1.2) Proof. Given two infinite sets \( \mathcal{P},{\mathcal{P}}^{\prime } \) of finite or infinite paths, let us \( \leq \) write \( \mathcal{P} \geq {\mathcal{P}}^{\prime } \) if \( {\mathcal{P}}^{\prime } \) consists of final segments of paths in \( \mathcal{P} \) . (Thus, if \( \mathcal{P} \) is a set of rays, then so is \( {\mathcal{P}}^{\prime } \) .) \( \omega \) Let \( G \) be any graph with a thick end \( \omega \) . Our task is to find disjoint rays in \( \omega \) that can serve as ’vertical’ (subdivided) rays \( {U}_{n} \) for our desired grid, and to link these up by suitable disjoint 'horizontal' paths. We begin by constructing a sequence \( {Q}_{0},{Q}_{1},\ldots \) of rays (of which we shall later choose some tails \( {Q}_{n}^{\prime } \) as ’vertical rays’), together with path systems \( \mathcal{P}\left( {Q}_{i}\right) \) between the \( {Q}_{i} \) and suitable \( {Q}_{p\left( i\right) } \) with \( p\left( i\right) < i \) (from which we  Fig. 8.2.1. The hexagonal quarter grid \( {H}^{\infty } \) . shall later choose the ’horizontal paths’). We shall aim to find the \( {Q}_{n} \) in ’supply sets’ \( {\mathcal{R}}_{0} \geq {\mathcal{R}}_{1} \geq \ldots \) of unused rays. We start with any infinite set \( {\mathcal{R}}_{0} \) of disjoint rays in \( \omega \) ; this exists by our assumption that \( \omega \) is a thick end. At step \( n \in \mathbb{N} \) of the construction, we shall choose the following: (1) a ray \( {Q}_{n} \in \omega \) disjoint from \( {Q}_{0} \cup \ldots \cup {Q}_{n - 1} \) ; (2) if \( n \geq 1 \), an integer \( p\left( n\right) < n \) ; (3) for every \( i \) with \( 1 \leq i \leq n \), an infinite set \( {\mathcal{P}}_{n}\left( {Q}_{i}\right) \) of disjoint \( {Q}_{i} - {Q}_{p\left( i\right) } \) paths, such that (i) \( \bigcup {\mathcal{P}}_{n}\left( {Q}_{i}\right) \cap \bigcup {\mathcal{P}}_{n}\left( {Q}_{j}\right) = \varnothing \) for distinct \( i, j \leq n \), and (ii) \( \bigcup {\mathcal{P}}_{n}\left( {Q}_{i}\right) \cap {Q}_{j} = \varnothing \) for distinct \( i, j \leq n \) with \( j \neq p\left( i\right) \) ; (4) an infinite set \( {\mathcal{R}}_{n + 1} \leq {\mathcal{R}}_{n} \) of disjoint rays that are disjoint from \( {Q}_{0} \cup \ldots \cup {Q}_{n} \) and from \( \bigcup {\mathcal{P}}_{n}\left( {Q}_{i}\right) \) whenever \( 1 \leq i \leq n \) . Thus, while the rays \( {Q}_{i} \) and the predecessor map \( i \mapsto p\left( i\right) \) remain unchanged once defined for some \( i \), the path system \( {\mathcal{P}}_{n}\left( {Q}_{i}\right) \) between \( {Q}_{i} \) and \( {Q}_{p\left( i\right) } \) changes as \( n \) increases. More precisely, we shall have (5) \( {\mathcal{P}}_{n}\left( {Q}_{i}\right) \subseteq {\mathcal{P}}_{n - 1}\left( {Q}_{i}\right) \) whenever \( 1 \leq i < n \) . Informally, we think of \( {\mathcal{P}}_{n}\left( {Q}_{i}\right) \) as our best candidate at time \( n \) for a system of horizontal paths linking \( {Q}_{i} \) to \( {Q}_{p\left( i\right) } \) . But, as new rays \( {Q}_{m} \) with \( m > n \) get selected, we may have to change our mind about \( {\mathcal{P}}_{n}\left( {Q}_{i}\right) \) and

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

1} \leq {\mathcal{R}}_{n} \) of disjoint rays that are disjoint from \( {Q}_{0} \cup \ldots \cup {Q}_{n} \) and from \( \bigcup {\mathcal{P}}_{n}\left( {Q}_{i}\right) \) whenever \( 1 \leq i \leq n \) . Thus, while the rays \( {Q}_{i} \) and the predecessor map \( i \mapsto p\left( i\right) \) remain unchanged once defined for some \( i \), the path system \( {\mathcal{P}}_{n}\left( {Q}_{i}\right) \) between \( {Q}_{i} \) and \( {Q}_{p\left( i\right) } \) changes as \( n \) increases. More precisely, we shall have (5) \( {\mathcal{P}}_{n}\left( {Q}_{i}\right) \subseteq {\mathcal{P}}_{n - 1}\left( {Q}_{i}\right) \) whenever \( 1 \leq i < n \) . Informally, we think of \( {\mathcal{P}}_{n}\left( {Q}_{i}\right) \) as our best candidate at time \( n \) for a system of horizontal paths linking \( {Q}_{i} \) to \( {Q}_{p\left( i\right) } \) . But, as new rays \( {Q}_{m} \) with \( m > n \) get selected, we may have to change our mind about \( {\mathcal{P}}_{n}\left( {Q}_{i}\right) \) and, again and again, prune it to a smaller system \( {\mathcal{P}}_{m}\left( {Q}_{i}\right) \) . This may leave us with an empty system at the end of of the construction. Thus, when we later come to construct our grid, we shall have to choose its horizontal paths between \( {Q}_{i} \) and \( {Q}_{p\left( i\right) } \) from these provisional sets \( {\mathcal{P}}_{n}\left( {Q}_{i}\right) \), not from their (possibly empty) intersection over all \( n \) . Let \( n \in \mathbb{N} \) be given. If \( n = 0 \), choose any ray from \( {\mathcal{R}}_{0} \) as \( {Q}_{0} \), and put \( {\mathcal{R}}_{1} \mathrel{\text{:=}} {\mathcal{R}}_{0} \smallsetminus \left\{ {Q}_{0}\right\} \) . Then conditions (1)-(5) hold for \( n = 0 \) . Suppose now that \( n \geq 1 \), and consider a ray \( {R}_{n}^{0} \in {\mathcal{R}}_{n} \) . By (4), \( {R}_{n}^{0} \) is disjoint from \[ H \mathrel{\text{:=}} {Q}_{0} \cup \ldots \cup {Q}_{n - 1} \cup \mathop{\bigcup }\limits_{{i = 1}}^{{n - 1}}{\mathcal{P}}_{n - 1}\left( {Q}_{i}\right) . \] By the choice of \( {\mathcal{R}}_{0} \) and (4), we know that \( {R}_{n}^{0} \in \omega \) . As also \( {Q}_{0} \in \omega \) , there exists an infinite set \( \mathcal{P} \) of disjoint \( {R}_{n}^{0} - H \) paths. If possible, we choose \( \mathcal{P} \) so that \( \bigcup \mathcal{P} \cap \bigcup {\mathcal{P}}_{n - 1}\left( {Q}_{i}\right) = \varnothing \) for all \( i \leq n - 1 \) . We may then further choose \( \mathcal{P} \) so that \( \bigcup \mathcal{P} \cap {Q}_{i} \neq \varnothing \) for only one \( i \), since by (1) the \( {Q}_{i} \) are disjoint for different \( i \) . We define \( p\left( n\right) \) as this \( i \), and put \( {\mathcal{P}}_{n}\left( {Q}_{j}\right) \mathrel{\text{:=}} {\mathcal{P}}_{n - 1}\left( {Q}_{j}\right) \) for all \( j \leq n - 1 \) . If \( \mathcal{P} \) cannot be chosen in this way, we may choose it so that all its vertices in \( H \) lie in \( \bigcup {\mathcal{P}}_{n - 1}\left( {Q}_{i}\right) \) for the same \( i \), since by (3) the graphs \( \bigcup {\mathcal{P}}_{n - 1}\left( {Q}_{i}\right) \) are disjoint for different \( i \) . We can then find infinite disjoint subsets \( {\mathcal{P}}_{n}\left( {Q}_{i}\right) \) of \( {\mathcal{P}}_{n - 1}\left( {Q}_{i}\right) \) and \( {\mathcal{P}}^{\prime } \) of \( \mathcal{P} \) . We continue infinitely many of the paths in \( {\mathcal{P}}^{\prime } \) along paths from \( {\mathcal{P}}_{n - 1}\left( {Q}_{i}\right) \smallsetminus {\mathcal{P}}_{n}\left( {Q}_{i}\right) \) to \( {Q}_{i} \) or to \( {Q}_{p\left( i\right) } \), to obtain an infinite set \( {\mathcal{P}}^{\prime \prime } \) of disjoint \( {R}_{n}^{0} - {Q}_{i} \) or \( {R}_{n}^{0} - {Q}_{p\left( i\right) } \) paths, and define \( p\left( n\right) \) as \( i \) or as \( p\left( i\right) \) accordingly. The paths in \( {\mathcal{P}}^{\prime \prime } \) then avoid \( \bigcup {\mathcal{P}}_{n}\left( {Q}_{j}\right) \) for all \( j \leq n - 1 \) (with \( {\mathcal{P}}_{n}\left( {Q}_{j}\right) \mathrel{\text{:=}} {\mathcal{P}}_{n - 1}\left( {Q}_{j}\right) \) for \( j \neq i \) ) and \( {Q}_{j} \) for all \( j \neq p\left( n\right) \) . We rename \( {\mathcal{P}}^{\prime \prime } \) as \( \mathcal{P} \), to simplify notation. \( {\mathcal{P}}_{n}\left( {Q}_{i}\right) \) In either case, we have now defined \( {\mathcal{P}}_{n}\left( {Q}_{i}\right) \) for all \( i < n \) so as to for \( i < n \) satisfy (5) for \( n \), chosen \( p\left( n\right) \) as in (2), and found an infinite set \( \mathcal{P} \) of \( p\left( n\right) ,\mathcal{P} \) disjoint \( {R}_{n}^{0} - {Q}_{p\left( n\right) } \) paths that avoid all other \( {Q}_{j} \) and all the sets \( {\mathcal{P}}_{n}\left( {Q}_{i}\right) \) . All that can prevent us from choosing \( {R}_{n}^{0} \) as \( {Q}_{n} \) and \( \mathcal{P} \) as \( {\mathcal{P}}_{n}\left( {Q}_{n}\right) \) and \( {\mathcal{R}}_{n + 1} \leq {\mathcal{R}}_{n} \smallsetminus \left\{ {R}_{n}^{0}\right\} \) is condition (4): if \( \mathcal{P} \) meets all but finitely many rays in \( {\mathcal{R}}_{n} \) infinitely, we cannot find an infinite set \( {\mathcal{R}}_{n + 1} \leq {\mathcal{R}}_{n} \) of rays avoiding \( \mathcal{P} \) . However, we may now assume the following: Whenever \( R \in {\mathcal{R}}_{n} \) and \( {\mathcal{P}}^{\prime } \leq \mathcal{P} \) is an infinite set of \( R - {Q}_{p\left( n\right) } \) \( \left( *\right) \) paths, there is a ray \( {R}^{\prime } \neq R \) in \( {\mathcal{R}}_{n} \) that meets \( {\mathcal{P}}^{\prime } \) infinitely. For if \( \left( *\right) \) failed, we could choose \( R \) as \( {Q}_{n} \) and \( {\mathcal{P}}^{\prime } \) as \( {\mathcal{P}}_{n}\left( {Q}_{n}\right) \), and select from every ray \( {R}^{\prime } \neq R \) in \( {\mathcal{R}}_{n} \) a tail avoiding \( {\mathcal{P}}^{\prime } \) to form \( {\mathcal{R}}_{n + 1} \) . This would satisfy conditions (1)-(5) for \( n \) . Consider the paths in \( \mathcal{P} \) as linearly ordered by the natural order of their starting vertices on \( {R}_{n}^{0} \) . This induces an ordering on every \( {\mathcal{P}}^{\prime } \leq \mathcal{P} \) . If \( {\mathcal{P}}^{\prime } \) is a set of \( R - {Q}_{p\left( n\right) } \) paths for some ray \( R \), we shall call this ordering of \( {\mathcal{P}}^{\prime } \) compatible with \( R \) if the ordering it induces on the first vertices of its paths coincides with the natural ordering of those vertices on \( R \) . Using assumption \( \left( *\right) \), let us choose two sequences \( {R}_{n}^{0},{R}_{n}^{1},\ldots \) and \( {\mathcal{P}}^{0} \geq {\mathcal{P}}^{1} \geq \ldots \) such that every \( {R}_{n}^{k} \) is a tail of a ray in \( {\mathcal{R}}_{n} \) and each \( {\mathcal{P}}^{k} \) is an infinite set of \( {R}_{n}^{k} - {Q}_{p\left( n\right) } \) paths whose ordering is compatible with \( {R}_{n}^{k} \) . The first path of \( {\mathcal{P}}^{k} \) in this ordering will be denoted by \( {P}_{k} \), its \( {P}_{k} \) starting vertex on \( {R}_{n}^{k} \) by \( {v}_{k} \), and the path in \( {\mathcal{P}}^{k - 1} \) containing \( {P}_{k} \) by \( {P}_{k}^{ - } \) (Fig. 8.2.2). Clearly, \( {\mathcal{P}}_{0} \mathrel{\text{:=}} \mathcal{P} \) is as required for \( k = 0 \) ; put \( {P}_{0}^{ - } \mathrel{\text{:=}} {P}_{0} \) . For \( k \geq 1 \), we may use \( \left( *\right) \) with \( R \supseteq {R}_{n}^{k - 1} \) and \( {\mathcal{P}}^{\prime } = {\mathcal{P}}^{k - 1} \) to find in \( {\mathcal{R}}_{n} \) a ray \( {R}^{\prime } \nsupseteq {R}_{n}^{k - 1} \) that meets \( {\mathcal{P}}^{k - 1} \) infinitely but has a tail \( {R}_{n}^{k} \) avoiding the finite subgraph \( {P}_{0}^{ - } \cup \ldots \cup {P}_{k - 1}^{ - } \) . Let \( {P}_{k}^{ - } \) be a path in \( {\mathcal{P}}^{k - 1} \) that meets \( {R}_{n}^{k} \) and let \( v \) be its ’highest’ vertex on \( {R}_{n}^{k} \), that is, the last vertex of \( {R}_{n}^{k} \) in \( V\left( {P}_{k}^{ - }\right) \) . Replacing \( {R}_{n}^{k} \) with its tail \( v{R}_{n}^{k} \), we can arrange that \( {P}_{k}^{ - } \) has only the vertex \( v \) on \( {R}_{n}^{k} \) . Then \( {P}_{k} \mathrel{\text{:=}} v{P}_{k}^{ - } \) is an \( {R}_{n}^{k} - {Q}_{p\left( n\right) } \) path starting at \( {v}_{k} = v \) . We may now select an infinite set \( {\mathcal{P}}^{k} \leq {\mathcal{P}}^{k - 1} \) of \( {R}_{n}^{k} - {Q}_{p\left( n\right) } \) paths compatible with \( {R}_{n}^{k} \) and containing \( {P}_{k} \) is its first path.  Fig. 8.2.2. Constructing \( {Q}_{n} \) from condition \( \left( *\right) \) Since \( {P}_{k}^{ - } \) contains \( {v}_{k} \in {R}_{n}^{k} \) but \( {R}_{n}^{k} \cap {P}_{k - 1} = \varnothing \), we have \( {P}_{k}^{ - } \neq {P}_{k - 1} \) , so the \( {P}_{k} \) are all disjoint. For each \( k \), let \( {v}_{k + 1}^{ - } \) denote the starting vertex of \( {P}_{k + 1}^{ - } \) on \( {R}_{n}^{k} \), and put \( {R}_{n + 1}^{k} \mathrel{\text{:=}} {\mathring{v}}_{k + 1}^{ - }{R}_{n}^{k} \) . Then let \( {R}_{n + 1}^{k} \) \[ {Q}_{n} \mathrel{\text{:=}} {v}_{0}{R}_{n}^{0}{v}_{1}^{ - }{P}_{1}^{ - }{v}_{1}{R}_{n}^{1}{v}_{2}^{ - }{P}_{2}^{ - }{v}_{2}{R}_{n}^{2}\ldots \] \[ {\mathcal{P}}_{n}\left( {Q}_{n}\right) \mathrel{\text{:=}} \left\{ {{P}_{0},{P}_{1},{P}_{2},\ldots }\right\} \] \[ {\mathcal{R}}_{n + 1} \mathrel{\text{:=}} \left\{ {{R}_{n + 1}^{k} \mid k \in \mathbb{N}}\right\} . \] Let us check that these definitions satisfy (1)-(5) for \( n \) . We have already verified (2) and (5). For the disjointness requirements in (1) and (3), recall that \( {Q}_{n} \) and \( {\mathcal{P}}_{n}\left( {Q}_{n}\right) \) consist of segments of paths in \( {\mathcal{R}}_{n} \) and \( \mathcal{P} \) ; these are disjoint from \( {Q}_{i} \) and \( {\mathcal{P}}_{n}\left( {Q}_{i}\right) \) for all \( i < n \) by definition of \( \mathcal{P} \) and (4) for \( n - 1 \) (together with (5) for \( n \) ). For the disjointness requirement in (4) note

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

mathring{v}}_{k + 1}^{ - }{R}_{n}^{k} \) . Then let \( {R}_{n + 1}^{k} \) \[ {Q}_{n} \mathrel{\text{:=}} {v}_{0}{R}_{n}^{0}{v}_{1}^{ - }{P}_{1}^{ - }{v}_{1}{R}_{n}^{1}{v}_{2}^{ - }{P}_{2}^{ - }{v}_{2}{R}_{n}^{2}\ldots \] \[ {\mathcal{P}}_{n}\left( {Q}_{n}\right) \mathrel{\text{:=}} \left\{ {{P}_{0},{P}_{1},{P}_{2},\ldots }\right\} \] \[ {\mathcal{R}}_{n + 1} \mathrel{\text{:=}} \left\{ {{R}_{n + 1}^{k} \mid k \in \mathbb{N}}\right\} . \] Let us check that these definitions satisfy (1)-(5) for \( n \) . We have already verified (2) and (5). For the disjointness requirements in (1) and (3), recall that \( {Q}_{n} \) and \( {\mathcal{P}}_{n}\left( {Q}_{n}\right) \) consist of segments of paths in \( {\mathcal{R}}_{n} \) and \( \mathcal{P} \) ; these are disjoint from \( {Q}_{i} \) and \( {\mathcal{P}}_{n}\left( {Q}_{i}\right) \) for all \( i < n \) by definition of \( \mathcal{P} \) and (4) for \( n - 1 \) (together with (5) for \( n \) ). For the disjointness requirement in (4) note that \( {R}_{n + 1}^{k} \) does not meet \( {Q}_{n} \) or \( {\mathcal{P}}_{n}\left( {Q}_{n}\right) \) inside any path \( {P}_{j}^{ - } \) with \( j > k + 1 \), since these \( {P}_{j}^{ - } \) are proper final segments of \( {R}_{n}^{k} - {Q}_{p\left( n\right) } \) paths in \( {\mathcal{P}}^{k} \) . Since \( {R}_{n + 1}^{k} \) does not, by definition, meet \( {Q}_{n} \) or \( {\mathcal{P}}_{n}\left( {Q}_{n}\right) \) inside any path \( {P}_{j}^{ - } \) with \( j \leq k + 1 \), condition (4) holds for \( n \) . It remains to use our rays \( {Q}_{n} \), path systems \( {\mathcal{P}}_{n}\left( {Q}_{i}\right) \), and supply sets \( {\mathcal{R}}_{n} \) of rays to construct the desired grid. By the infinity lemma (8.1.2), there is a sequence \( {n}_{0} < {n}_{1} < {n}_{2} < \ldots \) such that either \( p\left( {n}_{i}\right) = {n}_{i - 1} \) for every \( i \geq 1 \) or \( p\left( {n}_{i}\right) = {n}_{0} \) for every \( i \geq 1 \) . We treat these two cases in turn. In the first case, let us assume for notational simplicity that \( {n}_{i} = i \) for all \( i \), i.e. discard any \( {Q}_{n} \) with \( n \notin \left\{ {{n}_{0},{n}_{1},\ldots }\right\} \) . Then for every \( i \geq 1 \) and every \( n \geq i \) we have an infinite set \( {\mathcal{P}}_{n}\left( {Q}_{i}\right) \) of disjoint \( {Q}_{i} - {Q}_{i - 1} \) paths. Our aim is to choose tails \( {Q}_{n}^{\prime } \) of our rays \( {Q}_{n} \) that will correspond to the vertical rays \( {U}_{n} \subseteq {H}^{\infty } \), and paths \( {S}_{1},{S}_{2},\ldots \) between the \( {Q}_{n}^{\prime } \) that will correspond to the horizontal edges \( {e}_{1},{e}_{2},\ldots \) of \( {H}^{\infty } \) . We shall find the paths \( {S}_{1},{S}_{2},\ldots \) inductively, choosing the \( {Q}_{n}^{\prime } \) as needed as we go along (but also in the order of increasing \( n \), starting with \( {Q}_{0}^{\prime } \mathrel{\text{:=}} {Q}_{0} \) ). At every step of the construction, we shall have selected only finitely many \( {S}_{k} \) and only finitely many \( {Q}_{n}^{\prime } \) . Let \( k \) and \( n \) be minimal such that \( {S}_{k} \) and \( {Q}_{n}^{\prime } \) are still undefined. We describe how to choose \( {S}_{k} \), and \( {Q}_{n}^{\prime } \) if the definition of \( {S}_{k} \) requires it. Let \( i \) be such that \( {e}_{k} \) joins \( {U}_{i - 1} \) to \( {U}_{i} \) in \( {H}^{\infty } \) . If \( i = n \), let \( {Q}_{n}^{\prime } \) be a tail of \( {Q}_{n} \) that avoids the finitely many paths \( {S}_{1},\ldots ,{S}_{k - 1} \) ; otherwise, \( {Q}_{i}^{\prime } \) has already been defined, and so has \( {Q}_{i - 1}^{\prime } \) . Now choose \( {S}_{k} \in {\mathcal{P}}_{n}\left( {Q}_{i}\right) \) ’high enough’ between \( {Q}_{i - 1}^{\prime } \) and \( {Q}_{i}^{\prime } \) to mirror the position of \( {e}_{k} \) in \( {H}^{\infty } \), and to avoid \( {S}_{1} \cup \ldots \cup {S}_{k - 1} \) . By (3)(ii), \( {S}_{k} \) will also avoid every other \( {Q}_{j}^{\prime } \) already defined. Since every \( {Q}_{n}^{\prime } \) is chosen so as to avoid all previously defined \( {S}_{k} \), and every \( {S}_{k} \) avoids all previously defined \( {Q}_{j}^{\prime } \) (except \( {Q}_{i - 1}^{\prime } \) and \( \left. {Q}_{i}^{\prime }\right) \), the \( {Q}_{n}^{\prime } \) and \( {S}_{k} \) are pairwise disjoint for all \( n, k \in \mathbb{N} \), except for the required incidences. Our construction thus yields the desired subdivision of \( {H}^{\infty } \) . It remains to treat the case that \( p\left( {n}_{i}\right) = {n}_{0} \) for all \( i \geq 1 \) . Let us rename \( {Q}_{{n}_{0}} \) as \( Q \), and \( {n}_{i} \) as \( i - 1 \) for \( i \geq 1 \) . Then our sets \( {\mathcal{P}}_{n}\left( {Q}_{i}\right) \) consist of disjoint \( {Q}_{i} - Q \) paths. We choose rays \( {Q}_{n}^{\prime } \subseteq {Q}_{n} \) and paths \( {S}_{k} \) inductively as before, except that \( {S}_{k} \) now consists of three parts: an initial segment from \( {\mathcal{P}}_{n}\left( {Q}_{i - 1}\right) \), followed by a middle segment from \( Q \), and a final segment from \( {\mathcal{P}}_{n}\left( {Q}_{i}\right) \) . Such \( {S}_{k} \) can again be found, since at every stage of the construction only a finite part of \( Q \) has been used. ## 8.3 Homogeneous and universal graphs Unlike finite graphs, infinite graphs offer the possibility to represent an entire graph property \( \mathcal{P} \) by just one specimen, a single graph that contains all the graphs in \( \mathcal{P} \) up to some fixed cardinality. Such graphs are called 'universal' for this property. More precisely, if \( \leq \) is a graph relation (such as the minor, topological minor, subgraph, or induced subgraph relation up to isomorphism), we call a countable graph \( {G}^{ * }\; \) universal in \( \mathcal{P}\;\left( {\text{for} \leq }\right) \) if \( {G}^{ * } \in \mathcal{P} \) and \( G \leq {G}^{ * } \) universal for every countable graph \( G \in \mathcal{P} \) . Is there a graph that is universal in the class of all countable graphs? Suppose a graph \( R \) has the following property: Whenever \( U \) and \( W \) are disjoint finite sets of vertices in \( R \) , there exists a vertex \( v \in R - U - W \) that is adjacent in \( R \) \( \left( *\right) \) to all the vertices in \( U \) but to none in \( W \) . Then \( R \) is universal even for the strongest of all graph relations, the induced subgraph relation. Indeed, in order to embed a given countable graph \( G \) in \( R \) we just map its vertices \( {v}_{1},{v}_{2},\ldots \) to \( R \) inductively, making sure that \( {v}_{n} \) gets mapped to a vertex \( v \in R \) adjacent to the images of all the neighbours of \( {v}_{n} \) in \( G\left\lbrack {{v}_{1},\ldots ,{v}_{n}}\right\rbrack \) but not adjacent to the image of any non-neighbour of \( {v}_{n} \) in \( G\left\lbrack {{v}_{1},\ldots ,{v}_{n}}\right\rbrack \) . Clearly, this map is an isomorphism between \( G \) and the subgraph of \( R \) induced by its image. Theorem 8.3.1. (Erdős and Rényi 1963) --- \( \left\lbrack {11.3.5}\right\rbrack \) --- There exists a unique countable graph \( R \) with property \( \left( *\right) \) . Proof. To prove existence, we construct a graph \( R \) with property \( \left( *\right) \) inductively. Let \( {R}_{0} \mathrel{\text{:=}} {K}^{1} \) . For all \( n \in \mathbb{N} \), let \( {R}_{n + 1} \) be obtained from \( {R}_{n} \) by adding for every set \( U \subseteq V\left( {R}_{n}\right) \) a new vertex \( v \) joined to all the vertices in \( U \) but to none outside \( U \) . (In particular, the new vertices form an independent set in \( \left. {{R}_{n + 1}\text{.}}\right) \) Clearly \( R \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{n \in \mathbb{N}}}{R}_{n} \) has property \( \left( *\right) \) . To prove uniqueness, let \( R = \left( {V, E}\right) \) and \( {R}^{\prime } = \left( {{V}^{\prime },{E}^{\prime }}\right) \) be two graphs with property \( \left( *\right) \), each given with a fixed vertex enumeration. We construct a bijection \( \varphi : V \rightarrow {V}^{\prime } \) in an infinite sequence of steps, defining \( \varphi \left( v\right) \) for one new vertex \( v \in V \) at each step. At every odd step we look at the first vertex \( v \) in the enumeration of \( V \) for which \( \varphi \left( v\right) \) has not yet been defined. Let \( U \) be the set of those of its neighbours \( u \) in \( R \) for which \( \varphi \left( u\right) \) has already been defined. This is a finite set. Using \( \left( *\right) \) for \( {R}^{\prime } \), find a vertex \( {v}^{\prime } \in {V}^{\prime } \) that is adjacent in \( {R}^{\prime } \) to all the vertices in \( \varphi \left( U\right) \) but to no other vertex in the image of \( \varphi \) (which, so far, is still a finite set). Put \( \varphi \left( v\right) \mathrel{\text{:=}} {v}^{\prime } \) . At even steps in the definition process we do the same thing with the roles of \( R \) and \( {R}^{\prime } \) interchanged: we look at the first vertex \( {v}^{\prime } \) in the enumeration of \( {V}^{\prime } \) that does not yet lie in the image of \( \varphi \), and set \( \varphi \left( v\right) = {v}^{\prime } \) for a vertex \( v \) that matches the adjacencies and non-adjacencies of \( {v}^{\prime } \) among the vertices for which \( \varphi \) (resp. \( {\varphi }^{-1} \) ) has already been defined. By our minimum choices of \( v \) and \( {v}^{\prime } \), the bijection gets defined on all of \( V \) and all of \( {V}^{\prime } \), and it is clearly an isomorphism. --- Rado graph --- The graph \( R \) in Theorem 8.3.1 is usually called the Rado graph, named after Richard Rado who gave one of its earliest explicit definitions. The method of constructing a bijection in alternating steps, as in the uniqueness part of the proof, is known as the back-and-forth technique. The Rado graph \( R \) is unique in another rather fascinating respect. We shall hear more about this in Chapter 11.3, but in a

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we look at the first vertex \( {v}^{\prime } \) in the enumeration of \( {V}^{\prime } \) that does not yet lie in the image of \( \varphi \), and set \( \varphi \left( v\right) = {v}^{\prime } \) for a vertex \( v \) that matches the adjacencies and non-adjacencies of \( {v}^{\prime } \) among the vertices for which \( \varphi \) (resp. \( {\varphi }^{-1} \) ) has already been defined. By our minimum choices of \( v \) and \( {v}^{\prime } \), the bijection gets defined on all of \( V \) and all of \( {V}^{\prime } \), and it is clearly an isomorphism. --- Rado graph --- The graph \( R \) in Theorem 8.3.1 is usually called the Rado graph, named after Richard Rado who gave one of its earliest explicit definitions. The method of constructing a bijection in alternating steps, as in the uniqueness part of the proof, is known as the back-and-forth technique. The Rado graph \( R \) is unique in another rather fascinating respect. We shall hear more about this in Chapter 11.3, but in a nutshell it is the following. If we generate a countably infinite random graph by admitting its pairs of vertices as edges independently with some fixed positive probability \( p \in \left( {0,1}\right) \), then with probability 1 the resulting graph --- 'the' random graph --- has property \( \left( *\right) \), and is hence isomorphic to \( R \) ! In the context of infinite graphs, the Rado graph is therefore also called the (countably infinite) random graph. As one would expect of a random graph, the Rado graph shows a high degree of uniformity. One aspect of this is its resilience against small changes: the deletion of finitely many vertices or edges, and similar local changes, leave it ’unchanged’ and result in just another copy of \( R \) (Exercise 41). The following rather extreme aspect of uniformity, however, is still surprising: no matter how we partition the vertex set of \( R \) into two parts, at least one of the parts will induce another isomorphic copy of \( R \) . Trivial examples aside, the Rado graph is the only countable graph with this property, and hence unique in yet another respect: Proposition 8.3.2. The Rado graph is the only countable graph \( G \) other than \( {K}^{{\aleph }_{0}} \) and \( \overline{{K}^{{\aleph }_{0}}} \) such that, no matter how \( V\left( G\right) \) is partitioned into two parts, one of the parts induces an isomorphic copy of \( G \) . Proof. We first show that the Rado graph \( R \) has the partition property. Let \( \left\{ {{V}_{1},{V}_{2}}\right\} \) be a partition of \( V\left( R\right) \) . If \( \left( *\right) \) fails in both \( R\left\lbrack {V}_{1}\right\rbrack \) and \( R\left\lbrack {V}_{2}\right\rbrack \), say for sets \( {U}_{1},{W}_{1} \) and \( {U}_{2},{W}_{2} \), respectively, then \( \left( *\right) \) fails for \( U = {U}_{1} \cup {U}_{2} \) and \( W = {W}_{1} \cup {W}_{2} \) in \( R \), a contradiction. To show uniqueness, let \( G = \left( {V, E}\right) \) be a countable graph with the partition property. Let \( {V}_{1} \) be its set of isolated vertices, and \( {V}_{2} \) the rest. If \( {V}_{1} \neq \varnothing \) then \( G ≄ G\left\lbrack {V}_{2}\right\rbrack \), since \( G \) has isolated vertices but \( G\left\lbrack {V}_{2}\right\rbrack \) does not. Hence \( G = G\left\lbrack {V}_{1}\right\rbrack \simeq \overline{{K}^{{\aleph }_{0}}} \) . Similarly, if \( G \) has a vertex adjacent to all other vertices, then \( G = {K}^{{\aleph }_{0}} \) . Assume now that \( G \) has no isolated vertex and no vertex joined to all other vertices. If \( G \) is not the Rado graph then there are sets \( U, W \) for which (*) fails in \( G \) ; choose these with \( \left| {U \cup W}\right| \) minimum. Assume first that \( U \neq \varnothing \), and pick \( u \in U \) . Let \( {V}_{1} \) consist of \( u \) and all vertices outside \( U \cup W \) that are not adjacent to \( u \), and let \( {V}_{2} \) contain the remaining vertices. As \( u \) is isolated in \( G\left\lbrack {V}_{1}\right\rbrack \), we have \( G ≄ G\left\lbrack {V}_{1}\right\rbrack \) and hence \( G \simeq G\left\lbrack {V}_{2}\right\rbrack \) . By the minimality of \( \left| {U \cup W}\right| \), there is a vertex \( v \in G\left\lbrack {V}_{2}\right\rbrack - U - W \) that is adjacent to every vertex in \( U \smallsetminus \{ u\} \) and to none in \( W \) . But \( v \) is also adjacent to \( u \), because it lies in \( {V}_{2} \) . So \( U, W \) and \( v \) satisfy \( \left( *\right) \) for \( G \), contrary to assumption. Finally, assume that \( U = \varnothing \) . Then \( W \neq \varnothing \) . Pick \( w \in W \), and consider the partition \( \left\{ {{V}_{1},{V}_{2}}\right\} \) of \( V \) where \( {V}_{1} \) consists of \( w \) and all its neighbours outside \( W \) . As before, \( G ≄ G\left\lbrack {V}_{1}\right\rbrack \) and hence \( G \simeq G\left\lbrack {V}_{2}\right\rbrack \) . Therefore \( U \) and \( W \smallsetminus \{ w\} \) satisfy \( \left( *\right) \) in \( G\left\lbrack {V}_{2}\right\rbrack \), with \( v \in {V}_{2} \smallsetminus W \) say, and then \( U, W, v \) satisfy \( \left( *\right) \) in \( G \) . Another indication of the high degree of uniformity in the structure of the Rado graph is its large automorphism group. For example, \( R \) is easily seen to be vertex-transitive: given any two vertices \( x \) and \( y \), there is an automorphism of \( R \) mapping \( x \) to \( y \) . In fact, much more is true: using the back-and-forth technique, one --- homogeneous --- can easily show that the Rado graph is homogeneous: every isomorphism between two finite induced subgraphs can be extended to an automorphism of the entire graph (Exercise 42). Which other countable graphs are homogeneous? The complete graph \( {K}^{{\aleph }_{0}} \) and its complement are again obvious examples. Moreover, for every integer \( r \geq 3 \) there is a homogeneous \( {K}^{r} \) -free graph \( {R}^{r} \), constructed as follows. Let \( {R}_{0}^{r} \mathrel{\text{:=}} {K}^{1} \), and let \( {R}_{n + 1}^{r} \) be obtained from \( {R}_{n}^{r} \) by joining, for every subgraph \( H ≄ {K}^{r - 1} \) of \( {R}_{n}^{r} \), a new vertex \( {v}_{H} \) to every vertex in \( H \) . Then let \( {R}^{r} \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{n \in \mathbb{N}}}{R}_{n}^{r} \) . Clearly, as the new vertices \( {v}_{H} \) of \( {R}_{n + 1}^{r} \) are independent, there is no \( {K}^{r} \) in \( {R}_{n + 1}^{r} \) if there was none in \( {R}_{n}^{r} \), so \( {R}^{r} \nsupseteq {K}^{r} \) by induction on \( n \) . Just like the Rado graph, \( {R}^{r} \) is clearly universal among the \( {K}^{r} \) -free countable graphs, and it is clearly homogeneous. By the following deep theorem of Lachlan and Woodrow, the countable homogeneous graphs we have seen so far are essentially all: Theorem 8.3.3. (Lachlan & Woodrow 1980) Every countably infinite homogeneous graph is one of the following: - a disjoint union of complete graphs of the same order, or the complement of such a graph; - the graph \( {R}^{r} \) or its complement, for some \( r \geq 3 \) ; - the Rado graph \( R \) . To conclude this section, let us return to our original problem: for which graph properties is there a graph that is universal with this property? Most investigations into this problem have addressed it from a more general model-theoretic point of view, and have therefore been based on the strongest of all graph relations, the induced subgraph relation. Unfortunately, most of these results are negative; see the notes. From a graph-theoretic point of view, it seems more promising to look instead for universal graphs for the weaker subgraph relation, or even the topological minor or minor relation. For example, while there is no universal planar graph for subgraphs or induced subgraphs, there is one for minors: Theorem 8.3.4. (Diestel & Kühn 1999) There exists a universal planar graph for the minor relation. So far, this theorem is the only one of its kind. But it should be possible to find more. For instance: for which graphs \( X \) is there a minor-universal graph in the class \( {\operatorname{Forb}}_{ \preccurlyeq }\left( X\right) = \{ G \mid X \npreceq G\} \) ? ## 8.4 Connectivity and matching In this section we look at infinite versions of Menger's theorem and of the matching theorems from Chapter 2. This area of infinite graph theory is one of its best developed fields, with several deep results. One of these, however, stands out among the rest: a version of Menger's theorem that had been conjectured by Erdős and was proved only recently by Aharoni and Berger. The techniques developed for its proof inspired, over the years, much of the theory in this area. We shall prove this theorem for countable graphs, which will take up most of this section. Although the countable case is much easier, it is still quite hard and will give a good impression of the general proof. We then wind up with an overview of infinite matching theorems and a conjecture conceived in the same spirit. Recall that Menger’s theorem, in its simplest form, says that if \( A \) and \( B \) are sets of vertices in a finite graph \( G \), not necessarily disjoint, and if \( k = k\left( {G, A, B}\right) \) is the minimum number of vertices separating \( A \) from \( B \) in \( G \), then \( G \) contains \( k \) disjoint \( A - B \) paths. (Clearly, it cannot contain more.) The same holds, and is easily deduced from the finite case, when \( G \) is infinite but \( k \) is still finite: Proposition 8.4.1. Let \( G \) be any graph, \( k \in \mathbb{N} \), and let \( A, B \) be two sets of vertices in \( G \) that can be separated by \( k \) but no fewer than \( k \) vertices. Then \( G \) contains \( k \) disjoint \( A - B \) paths. (3.3.1) Proof. By assumption, every set of disjoint \( A - B \) paths has cardinality at most \( k \) . Choose one, \( \mathcal{P} \) say, of maximum cardinality. Suppose \( \left| \mathcal{P}\right| < k \) . Then no

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a conjecture conceived in the same spirit. Recall that Menger’s theorem, in its simplest form, says that if \( A \) and \( B \) are sets of vertices in a finite graph \( G \), not necessarily disjoint, and if \( k = k\left( {G, A, B}\right) \) is the minimum number of vertices separating \( A \) from \( B \) in \( G \), then \( G \) contains \( k \) disjoint \( A - B \) paths. (Clearly, it cannot contain more.) The same holds, and is easily deduced from the finite case, when \( G \) is infinite but \( k \) is still finite: Proposition 8.4.1. Let \( G \) be any graph, \( k \in \mathbb{N} \), and let \( A, B \) be two sets of vertices in \( G \) that can be separated by \( k \) but no fewer than \( k \) vertices. Then \( G \) contains \( k \) disjoint \( A - B \) paths. (3.3.1) Proof. By assumption, every set of disjoint \( A - B \) paths has cardinality at most \( k \) . Choose one, \( \mathcal{P} \) say, of maximum cardinality. Suppose \( \left| \mathcal{P}\right| < k \) . Then no set \( X \) consisting of one vertex from each path in \( \mathcal{P} \) separates \( A \) from \( B \) . For each \( X \), let \( {P}_{X} \) be an \( A - B \) path avoiding \( X \) . Let \( H \) be the union of \( \bigcup \mathcal{P} \) with all these paths \( {P}_{X} \) . This is a finite graph in which no set of \( \left| \mathcal{P}\right| \) vertices separates \( A \) from \( B \) . So \( H \subseteq G \) contains more than \( \left| \mathcal{P}\right| \) paths from \( A \) to \( B \) by Menger’s theorem (3.3.1), which contradicts the choice of \( \mathcal{P} \) . When \( k \) is infinite, however, the result suddenly becomes trivial. Indeed, let \( \mathcal{P} \) be any maximal set of disjoint \( A - B \) paths in \( G \) . Then the union of all these paths separates \( A \) from \( B \), so \( \mathcal{P} \) must be infinite. But then the cardinality of this union is no bigger than \( \left| \mathcal{P}\right| \) . Thus, \( \mathcal{P} \) contains \( \left| \mathcal{P}\right| = \left| {\bigcup \mathcal{P}}\right| \geq k \) disjoint \( A - B \) paths, as desired. Of course, this is no more than a trick played on us by infinite cardinal arithmetic: although, numerically, the \( A - B \) separator consisting of all the inner vertices of paths in \( \mathcal{P} \) is no bigger than \( \left| \mathcal{P}\right| \), it uses far more vertices to separate \( A \) from \( B \) than should be necessary. Or put another way: when our path systems and separators are infinite, their cardinalities alone are no longer a sufficiently fine tool to distinguish carefully chosen 'small' separators from unnecessarily large and wasteful ones. To overcome this problem, Erdős suggested an alternative form of Menger's theorem, which for finite graphs is clearly equivalent to the standard version. Recall that an \( A - B \) separator \( X \) is said to lie on a set \( \mathcal{P} \) of disjoint \( A - B \) paths if \( X \) consists of a choice of exactly one vertex from --- Erdős-Menger conjecture --- each path in \( \mathcal{P} \) . The following so-called Erdős-Menger conjecture, now a theorem, influenced much of the development of infinite connectivity and matching theory: Theorem 8.4.2. (Aharoni & Berger 2005) Let \( G \) be any graph, and let \( A, B \subseteq V\left( G\right) \) . Then \( G \) contains a set \( \mathcal{P} \) of disjoint \( A - B \) paths and an \( A - B \) separator on \( \mathcal{P} \) . The next few pages give a proof of Theorem 8.4.2 for countable \( G \) . Of the three proofs we gave for the finite case of Menger's theorem, only the last has any chance of being adaptable to the infinite case: the others were by induction on \( \left| \mathcal{P}\right| \) or on \( \left| G\right| + \parallel G\parallel \), and both these parameters may now be infinite. The third proof, however, looks more promising: recall that, by Lemmas 3.3.2 and 3.3.3, it provided us with a tool to either find a separator on a given system of \( A - B \) paths, or to construct another system of \( A - B \) paths that covers more vertices in \( A \) and in \( B \) . Lemmas 3.3.2 and 3.3.3 (whose proofs work for infinite graphs too) will indeed form a cornerstone of our proof for Theorem 8.4.2. However, it will not do just to apply these lemmas infinitely often. Indeed, although any finite number of applications of Lemma 3.3.2 leaves us with another system of disjoint \( A - B \) paths, an infinite number of iterations may leave nothing at all: each edge may be toggled on and off infinitely often by successive alternating paths, so that no ’limit system’ of \( A - B \) paths will be defined. We shall therefore take another tack: starting at \( A \) , we grow simultaneously as many disjoint paths towards \( B \) as possible. To make this precise, we need some terminology. Given a set \( X \subseteq \) \( V\left( G\right) \), let us write \( {G}_{X \rightarrow B} \) for the subgraph of \( G \) induced by \( X \) and all \( {G}_{X \rightarrow B} \) the components of \( G - X \) that meet \( B \) . Let \( \mathcal{W} = \left( {{W}_{a} \mid a \in A}\right) \) be a family of disjoint paths such that every \( {W}_{a} \) starts in \( a \) . We call \( \mathcal{W} \) an \( A \rightarrow B \) wave in \( G \) if the set \( Z \) of final vertices of paths in \( \mathcal{W} \) separates \( A \) from \( B \) in \( G \) . (Note that \( \mathcal{W} \) may contain infinite paths, which have no final vertex.) Sometimes, we shall wish to consider \( A \rightarrow B \) waves in subgraphs of \( G \) that contain \( A \) but not all of \( B \) . For this reason we do not formally require that \( B \subseteq V\left( G\right) \) .  Fig. 8.4.1. A small \( A \rightarrow B \) wave \( \mathcal{W} \) with boundary \( X \) When \( \mathcal{W} \) is a wave, then the set \( X \subseteq Z \) of those vertices in \( Z \) that either lie in \( B \) or have a neighbour in \( {G}_{Z \rightarrow B} - Z \) is a minimal \( A - B \) separator in \( G \) ; note that \( z \in Z \) lies in \( X \) if and only if it can be --- boundary \( \left( {\mathcal{W}, X}\right) \) --- linked to \( B \) by a path that has no vertex other than \( z \) on \( \mathcal{W} \) . We call \( X \) the boundary of \( \mathcal{W} \), and often use \( \left( {\mathcal{W}, X}\right) \) as shorthand for the wave \( \mathcal{W} \) together with its boundary \( X \) . If all the paths in \( \mathcal{W} \) are finite and --- large/small proper --- \( X = Z \), we call the wave \( \mathcal{W} \) large; otherwise it is small. We shall call \( \mathcal{W} \) proper if at least one of the paths in \( \mathcal{W} \) is non-trivial, or if all its paths are trivial but its boundary is a proper subset of \( A \) . Every small wave, for example, is proper. Note that while some \( A \rightarrow B \) wave always exists, e.g. the family \( \left( {\{ a\} \mid a \in A}\right) \) of singleton paths, \( G \) need not have a proper \( A \rightarrow B \) wave. (For example, if \( A \) consists of two vertices of \( G = {K}^{10} \) and \( B \) of three other vertices, there is no proper \( A \rightarrow B \) wave.) If \( \left( {\mathcal{U}, X}\right) \) is an \( A \rightarrow B \) wave in \( G \) and \( \left( {\mathcal{V}, Y}\right) \) is an \( X \rightarrow B \) wave \( \mathcal{U} + \mathcal{V} \) in \( {G}_{X \rightarrow B} \), then the family \( \mathcal{W} = \mathcal{U} + \mathcal{V} \) obtained from \( \mathcal{U} \) by appending the paths of \( \mathcal{V} \) (to those paths of \( \mathcal{U} \) that end in \( X \) ) is clearly an \( A \rightarrow B \) wave in \( G \), with boundary \( Y \) . Note that \( \mathcal{W} \) is large if and only if both \( \mathcal{V} \) and \( \mathcal{U} \) are large. \( \mathcal{W} \) is greater than \( \mathcal{U} \) in the following sense. Given two path systems \( \mathcal{U} = \left( {{U}_{a} \mid a \in A}\right) \) and \( \mathcal{W} = \left( {{W}_{a} \mid a \in A}\right) \) , \( \leq \) write \( \mathcal{U} \leq \mathcal{W} \) if \( {U}_{a} \subseteq {W}_{a} \) for every \( a \in A \) . Given a chain \( {\left( {\mathcal{W}}^{i},{X}^{i}\right) }_{i \in I} \) of waves in this ordering, with \( {\mathcal{W}}^{i} = \left( {{W}_{a}^{i} \mid a \in A}\right) \) say, let \( {\mathcal{W}}^{ * } = \left( {{W}_{a}^{ * } \mid }\right. \) \( a \in A \) ) be defined by \( {W}_{a}^{ * } \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{i \in I}}{W}_{a}^{i} \) . Then \( {\mathcal{W}}^{ * } \) is an \( A \rightarrow B \) wave: any \( A - B \) path is finite but meets every \( {X}^{i} \), so at least one of its vertices lies in \( {X}^{i} \) for arbitrarily large \( \left( {{\mathcal{W}}^{i},{X}^{i}}\right) \) and hence is the final vertex of limit wave a path in \( {\mathcal{W}}^{ * } \) . Clearly \( {\mathcal{W}}^{i} \leq {\mathcal{W}}^{ * } \) for all \( i \in I \) ; we call \( {\mathcal{W}}^{ * } \) the limit of the waves \( {\mathcal{W}}^{i} \) . As every chain of \( A \rightarrow B \) waves is bounded above by its limit wave, maximal Zorn’s lemma implies that \( G \) has a maximal \( A \rightarrow B \) wave \( \mathcal{W} \) ; let \( X \) be wave its boundary. This wave \( \left( {\mathcal{W}, X}\right) \) forms the first step in our proof for Theorem 8.4.2: if we can now find disjoint paths in \( {G}_{X \rightarrow B} \) linking all the vertices of \( X \) to \( B \), then \( X \) will be an \( A - B \) separator on these paths preceded by the paths of \( \mathcal{W} \) that end in \( X \) . By the maximality of \( \mathcal{W} \), there is no proper \( X \rightarrow B \) wave in \( {G}_{X \rightarrow B} \) . For our proof it will thus suffice to prove the following (renaming \( X \) as \( A \) ): Lemma 8.4.3. If \( G \) has no proper \( A \rightarrow B \) wave, then \( G \) contains a set of disjoint \( A - B \) paths linking all of \( A \) to \( B \) . Our approach to the

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the limit of the waves \( {\mathcal{W}}^{i} \) . As every chain of \( A \rightarrow B \) waves is bounded above by its limit wave, maximal Zorn’s lemma implies that \( G \) has a maximal \( A \rightarrow B \) wave \( \mathcal{W} \) ; let \( X \) be wave its boundary. This wave \( \left( {\mathcal{W}, X}\right) \) forms the first step in our proof for Theorem 8.4.2: if we can now find disjoint paths in \( {G}_{X \rightarrow B} \) linking all the vertices of \( X \) to \( B \), then \( X \) will be an \( A - B \) separator on these paths preceded by the paths of \( \mathcal{W} \) that end in \( X \) . By the maximality of \( \mathcal{W} \), there is no proper \( X \rightarrow B \) wave in \( {G}_{X \rightarrow B} \) . For our proof it will thus suffice to prove the following (renaming \( X \) as \( A \) ): Lemma 8.4.3. If \( G \) has no proper \( A \rightarrow B \) wave, then \( G \) contains a set of disjoint \( A - B \) paths linking all of \( A \) to \( B \) . Our approach to the proof of Lemma 8.4.3 is to enumerate the vertices in \( A = : \left\{ {{a}_{1},{a}_{2},\ldots }\right\} \), and to find the required \( A - B \) paths \( {P}_{n} = \) --- \( {a}_{1},{a}_{2},\ldots \) --- \( {a}_{n}\ldots {b}_{n} \) in turn for \( n = 1,2,\ldots \) . Since our premise in Lemma 8.4.3 is that \( G \) has no proper \( A \rightarrow B \) wave, we would like to choose \( {P}_{1} \) so that \( G - {P}_{1} \) has no proper \( \left( {A \smallsetminus \left\{ {a}_{1}\right\} }\right) \rightarrow B \) wave: this would restore the same premise to \( G - {P}_{1} \), and we could proceed to find \( {P}_{2} \) in \( G - {P}_{1} \) in the same way. We shall not be able to choose \( {P}_{1} \) just like this, but we shall be able to do something almost as good. We shall construct \( {P}_{1} \) so that deleting it (as well as a few more vertices outside \( A \) ) leaves a graph that has a large maximal \( \left( {A \smallsetminus \left\{ {a}_{1}\right\} }\right) \rightarrow B \) wave \( \left( {\mathcal{W},{A}^{\prime }}\right) \) . We then earmark the paths \( {W}_{n} = {a}_{n}\ldots {a}_{n}^{\prime }\left( {n \geq 2}\right) \) of this wave as initial segments for the paths \( {P}_{n} \) . By the maximality of \( \mathcal{W} \), there is no proper \( {A}^{\prime } \rightarrow B \) wave in \( {G}_{{A}^{\prime } \rightarrow B} \) . In other words, we have restored our original premise to \( {G}_{{A}^{\prime } \rightarrow B} \) , and can find there an \( {A}^{\prime } - B \) path \( {P}_{2}^{\prime } = {a}_{2}^{\prime }\ldots {b}_{2} \) . Then \( {P}_{2} \mathrel{\text{:=}} {a}_{2}{W}_{2}{a}_{2}^{\prime }{P}_{2}^{\prime } \) is our second path for Lemma 8.4.3, and we continue inductively inside \( {G}_{{A}^{\prime } \rightarrow B} \) . Given a set \( \widehat{A} \) of vertices in \( G \), let us call a vertex \( a \notin \widehat{A} \) linkable linkable for \( \left( {G,\widehat{A}, B}\right) \) if \( G - \widehat{A} \) contains an \( a - B \) path \( P \) and a set \( X \supseteq V\left( P\right) \) of vertices such that \( G - X \) has a large maximal \( \widehat{A} \rightarrow B \) wave. (The first such \( a \) we shall be considering will be \( {a}_{1} \), and \( \widehat{A} \) will be the set \( \left. {\left\{ {{a}_{2},{a}_{3},\ldots }\right\} \text{. }}\right) \) Lemma 8.4.4. Let \( {a}^{ * } \in A \) and \( \widehat{A} \mathrel{\text{:=}} A \smallsetminus \left\{ {a}^{ * }\right\} \), and assume that \( G \) has no proper \( A \rightarrow B \) wave. Then \( {a}^{ * } \) is linkable for \( \left( {G,\widehat{A}, B}\right) \) . Proof of Lemma 8.4.3 (assuming Lemma 8.4.4). Let \( G \) be as in Lemma 8.4.3, i.e. assume that \( G \) has no proper \( A \rightarrow B \) wave. We construct subgraphs \( {G}_{1},{G}_{2},\ldots \) of \( G \) satisfying the following statement (Fig. 8.4.2): \( {G}_{n} \) contains a set \( {A}^{n} = \left\{ {{a}_{n}^{n},{a}_{n + 1}^{n},{a}_{n + 2}^{n},\ldots }\right\} \) of distinct vertices such that \( {G}_{n} \) has no proper \( {A}^{n} \rightarrow B \) wave. In \( G \) there are disjoint paths \( {P}_{i}\left( {i < n}\right) \) and \( {W}_{i}^{n}\left( {i \geq n}\right) \) \( \left( *\right) \) starting at \( {a}_{i} \) . The \( {P}_{i} \) are disjoint from \( {G}_{n} \) and end in \( B \) . The \( {W}_{i}^{n} \) end in \( {a}_{i}^{n} \) and are otherwise disjoint from \( {G}_{n} \) . Clearly, the paths \( {P}_{1},{P}_{2},\ldots \) will satisfy Lemma 8.4.3.  Fig. 8.4.2. \( {G}_{n} \) has no proper \( {A}^{n} \rightarrow B \) wave Let \( {G}_{1} \mathrel{\text{:=}} G \), and put \( {a}_{i}^{1} \mathrel{\text{:=}} {a}_{i} \) and \( {W}_{i}^{1} \mathrel{\text{:=}} \left\{ {a}_{i}\right\} \) for all \( i \geq 1 \) . Since by assumption \( G \) has no proper \( A \rightarrow B \) wave, these definitions satisfy \( \left( *\right) \) for \( n = 1 \) . Suppose now that \( \left( *\right) \) has been satisfied for \( n \) . Put \( {\widehat{A}}^{n} \mathrel{\text{:=}} {A}^{n} \smallsetminus \left\{ {a}_{n}^{n}\right\} \) . By Lemma 8.4.4 applied to \( {G}_{n} \), we can find in \( {G}_{n} - {\widehat{A}}^{n} \) an \( {a}_{n}^{n} - B \) path \( P \) and a set \( {X}_{n} \supseteq V\left( P\right) \) such that \( {G}_{n} - {X}_{n} \) has a large maximal \( {\widehat{A}}^{n} \rightarrow B \) wave \( \left( {\mathcal{W},{A}^{n + 1}}\right) \) . Let \( {P}_{n} \) be the path \( {W}_{n}^{n} \cup P \) . For \( i \geq n + 1 \), let \( {W}_{i}^{n + 1} \) be \( {W}_{i}^{n} \) followed by the path of \( \mathcal{W} \) starting at \( {a}_{i}^{n} \), and call its last vertex \( {a}_{i}^{n + 1} \) . By the maximality of \( \mathcal{W} \) there is no proper \( {A}^{n + 1} \rightarrow B \) wave in \( {G}_{n + 1} \mathrel{\text{:=}} {\left( {G}_{n} - {X}_{n}\right) }_{{A}^{n + 1} \rightarrow B} \), so \( \left( *\right) \) is satisfied for \( n + 1 \) . To complete our proof of Theorem 8.4.2, it remains to prove Lemma 8.4.4. For this, we need another lemma: Lemma 8.4.5. Let \( x \) be a vertex in \( G - A \) . If \( G \) has no proper \( A \rightarrow B \) wave but \( G - x \) does, then every \( A \rightarrow B \) wave in \( G - x \) is large. \( \left( {3.3.2}\right) \) Proof. Suppose \( G - x \) has a small \( A \rightarrow B \) wave \( \left( {\mathcal{W}, X}\right) \) . Put \( {B}^{\prime } \mathrel{\text{:=}} \) \( X \cup \{ x\} \), and let \( \mathcal{P} \) denote the set of \( A - X \) paths in \( \mathcal{W} \) (Fig. 8.4.3). If \( G \) contains an \( A - {B}^{\prime } \) separator \( S \) on \( \mathcal{P} \), then replacing in \( \mathcal{W} \) every \( P \in \mathcal{P} \)  Fig. 8.4.3. A hypothetical small \( A \rightarrow B \) wave in \( G - x \) with its initial segment ending in \( S \) we obtain a small (and hence proper) \( A \rightarrow B \) wave in \( G \), which by assumption does not exist. By Lemmas 3.3.3 and 3.3.2, therefore, \( G \) contains a set \( {\mathcal{P}}^{\prime } \) of disjoint \( A - {B}^{\prime } \) paths exceeding \( \mathcal{P} \) . The set of last vertices of these paths contains \( X \) properly, and hence must be all of \( {B}^{\prime } = X \cup \{ x\} \) . But \( {B}^{\prime } \) separates \( A \) from \( B \) in \( G \), so we can turn \( {\mathcal{P}}^{\prime } \) into an \( A \rightarrow B \) wave in \( G \) by adding as singleton paths any vertices of \( A \) it does not cover. As \( x \) lies on \( {\mathcal{P}}^{\prime } \) but not in \( A \) , this is a proper wave, which by assumption does not exist. Proof of Lemma 8.4.4. We inductively construct trees \( {T}_{0} \subseteq {T}_{1} \subseteq \ldots \) in \( G - \left( {\widehat{A} \cup B}\right) \) and path systems \( {\mathcal{W}}_{0} \leq {\mathcal{W}}_{1} \leq \ldots \) in \( G \) so that each \( {\mathcal{W}}_{n} \) \( {\mathcal{W}}_{n} \) is a large maximal \( \widehat{A} \rightarrow B \) wave in \( G - {T}_{n} \) . Let \( {\mathcal{W}}_{0} \mathrel{\text{:=}} \left( {\{ a\} \mid a \in \widehat{A}}\right) \) . Clearly, \( {\mathcal{W}}_{0} \) is an \( \widehat{A} \rightarrow B \) wave in \( G - {a}^{ * } \) , and it is large and maximal: if not, then \( G - {a}^{ * } \) has a proper \( \widehat{A} \rightarrow B \) wave, and adding the trivial path \( \left\{ {a}^{ * }\right\} \) to this wave turns it into a proper \( A \rightarrow B \) wave (which by assumption does not exist). If \( {a}^{ * } \in B \) , the existence of \( {\mathcal{W}}_{0} \) makes \( {a}^{ * } \) linkable for \( \left( {G,\widehat{A}, B}\right) \) . So we assume that \( {a}^{ * } \notin B \) . Now \( {T}_{0} \mathrel{\text{:=}} \left\{ {a}^{ * }\right\} \) and \( {\mathcal{W}}_{0} \) are as desired. Suppose now that \( {T}_{n} \) and \( {\mathcal{W}}_{n} \) have been defined, and let \( {A}_{n} \) denote \( {A}_{n} \) the set of last vertices of the paths in \( {\mathcal{W}}_{n} \) . Since \( {\mathcal{W}}_{n} \) is large, \( {A}_{n} \) is its boundary, and since \( {\mathcal{W}}_{n} \) is maximal, \( {G}_{n} \mathrel{\text{:=}} {\left( G - {T}_{n}\right) }_{{A}_{n} \rightarrow B} \) has no proper \( {G}_{n} \) \( {A}_{n} \rightarrow B \) wave (Fig. 8.4.4).  Fig. 8.4.4. As \( {\mathcal{W}}_{n} \) is maximal, \( {G}_{n} \) has no proper \( {A}_{n} \rightarrow B \) wave Note that \( {A}_{n} \) does not separate \( A \) from \( B \) in \( G \) : if it did, then \( {\mathcal{W}}_{n} \cup \left\{ {a}^{ * }\right\} \) would be a small \( A \rightarrow B \) wave in \( G \), which does not exist. Hence, \( G - {A}_{n} \) contains an \( A - B \) path \( P \), which meets \( {T}_{n} \) because \( \left( {{\mathcal{W}}_{n},{A}_{n}}\right) \) is a wave in \( G - {T}_{n} \) . Let \( {P}_{n} \) be such a path \( P \), chosen so t

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

es of the paths in \( {\mathcal{W}}_{n} \) . Since \( {\mathcal{W}}_{n} \) is large, \( {A}_{n} \) is its boundary, and since \( {\mathcal{W}}_{n} \) is maximal, \( {G}_{n} \mathrel{\text{:=}} {\left( G - {T}_{n}\right) }_{{A}_{n} \rightarrow B} \) has no proper \( {G}_{n} \) \( {A}_{n} \rightarrow B \) wave (Fig. 8.4.4).  Fig. 8.4.4. As \( {\mathcal{W}}_{n} \) is maximal, \( {G}_{n} \) has no proper \( {A}_{n} \rightarrow B \) wave Note that \( {A}_{n} \) does not separate \( A \) from \( B \) in \( G \) : if it did, then \( {\mathcal{W}}_{n} \cup \left\{ {a}^{ * }\right\} \) would be a small \( A \rightarrow B \) wave in \( G \), which does not exist. Hence, \( G - {A}_{n} \) contains an \( A - B \) path \( P \), which meets \( {T}_{n} \) because \( \left( {{\mathcal{W}}_{n},{A}_{n}}\right) \) is a wave in \( G - {T}_{n} \) . Let \( {P}_{n} \) be such a path \( P \), chosen so that \( {P}_{n} \) its vertex \( {p}_{n} \) following its last vertex \( {t}_{n} \) in \( {T}_{n} \) is chosen minimal in some \( {p}_{n},{t}_{n} \) fixed enumeration of \( V\left( G\right) \) . Note that \( {p}_{n}{P}_{n} \subseteq {G}_{n} - {A}_{n} \), by definition of \( {G}_{n} \) . Now \( {P}_{n}^{\prime } = {a}^{ * }{T}_{n}{t}_{n}{P}_{n} \) is an \( {a}^{ * } - B \) path in \( G - \widehat{A} - {A}^{n} \) . If \( {G}_{n} - {p}_{n}{P}_{n} \) has no proper \( {A}_{n} \rightarrow B \) wave, then \( {\mathcal{W}}_{n} \) is large and maximal not only in \( G - {T}_{n} \) but also in \( G - {T}_{n} - {p}_{n}{P}_{n} \), and \( {a}^{ * } \) is linkable for \( \left( {G,\widehat{A}, B}\right) \) with \( {a}^{ * } - B \) path \( {P}_{n}^{\prime } \) and \( X = V\left( {{T}_{n} \cup {p}_{n}{P}_{n}}\right) \) . We may therefore assume that \( {G}_{n} - {p}_{n}{P}_{n} \) has a proper \( {A}_{n} \rightarrow B \) wave. Let \( {x}_{n} \) be the first vertex on \( {p}_{n}{P}_{n} \) such that \( {G}_{n} - {p}_{n}{P}_{n}{x}_{n} \) has a proper \( {A}_{n} \rightarrow B \) wave. Then \( {G}_{n}^{\prime } \mathrel{\text{:=}} {G}_{n} - {p}_{n}{P}_{n}{\mathring{x}}_{n} \) has no proper \( {A}_{n} \rightarrow B \) wave but \( {G}_{n}^{\prime } - {x}_{n} \) does, so by Lemma 8.4.5 every \( {A}_{n} \rightarrow B \) wave in \( {G}_{n}^{\prime } - {x}_{n} = {G}_{n} - {p}_{n}{P}_{n}{x}_{n} \) is large. Let \( \mathcal{W} \) be a maximal such wave, put \( {\mathcal{W}}_{n + 1} \mathrel{\text{:=}} {\mathcal{W}}_{n} + \mathcal{W} \), and let \( {T}_{n + 1} \mathrel{\text{:=}} {T}_{n} \cup {t}_{n}{P}_{n}{x}_{n} \) . Then \( {\mathcal{W}}_{n + 1} \) is a large maximal \( \widehat{A} \rightarrow B \) wave in \( G - {T}_{n + 1} \) . If \( {x}_{n} \in B \), then \( {T}_{n + 1} \) contains a path linking \( {a}^{ * } \) to \( B \), which satisfies the lemma with \( {\mathcal{W}}_{n + 1} \) and \( X = V\left( {T}_{n + 1}\right) \) . We may therefore assume that \( {x}_{n} \notin B \), giving \( {T}_{n + 1} \subseteq G - \left( {\widehat{A} \cup B}\right) \) as required. Put \( {T}^{ * } \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{n \in \mathbb{N}}}{T}_{n} \) . Then the \( {\mathcal{W}}_{n} \) are \( \widehat{A} \rightarrow B \) waves in \( G - {T}^{ * } \) ; let \( \left( {{\mathcal{W}}^{ * },{A}^{ * }}\right) \) be their limit. Our aim is to show that \( {A}^{ * } \) separates \( A \) from \( B \) not only in \( G - {T}^{ * } \) but even in \( G \) : then \( \left( {{\mathcal{W}}^{ * } \cup \left\{ {a}^{ * }\right\} ,{A}^{ * }}\right) \) is a small \( A \rightarrow B \) wave in \( G \), a contradiction. Suppose there exists an \( A - B \) path \( Q \) in \( G - {A}^{ * } \) . Let \( t \) be its last vertex in \( {T}^{ * } \) . Since \( {T}^{ * } \) does not meet \( B \), there is a vertex \( p \) following \( t \) on \( Q \) . Since \( {T}^{ * } \) contains every \( {p}_{n} \) but not \( p \), the path \( P = {a}^{ * }{T}^{ * }{tQ} \) was never chosen as \( {P}_{n} \) . Now let \( n \) be large enough that \( t \in {T}_{n} \), and that \( p \) precedes \( {p}_{n} \) in our fixed enumeration of \( V\left( G\right) \) . The fact that \( P \) was not chosen as \( {P}_{n} \) then means that its portion \( {pQ} \) outside \( {T}_{n} \) meets \( {A}_{n} \), say in a vertex \( q \) . Now \( q \notin {A}^{ * } \) by the choice of \( Q \) . Let \( W \) be the path in \( {\mathcal{W}}_{n} \) that joins \( \widehat{A} \) to \( q \) ; this path too avoids \( {A}^{ * } \) . But then \( {WqQ} \) contains an \( \widehat{A} - B \) path in \( G - {T}^{ * } \) avoiding \( {A}^{ * } \), which contradicts the definition of \( {A}^{ * } \) . The proof of Theorem 8.4.2 for countable \( G \) is now complete. Turning now to matching, let us begin with a simple problem that is intrinsically infinite. Given two sets \( A, B \) and injective functions \( A \rightarrow B \) and \( B \rightarrow A \), is there necessarily also a bijection between \( A \) and \( B \) ? Indeed there is - this is the famous Cantor-Bernstein theorem from elementary set theory. Recast in terms of matchings, the proof becomes very simple: Proposition 8.4.6. Let \( G \) be a bipartite graph, with bipartition \( \{ A, B\} \) say. If \( G \) contains a matching of \( A \) and a matching of \( B \), then \( G \) has a 1-factor. Proof. Let \( H \) be the multigraph on \( V\left( G\right) \) whose edge set is the disjoint union of the two matchings. (Thus, any edge that lies in both matchings becomes a double edge in \( H \) .) Every vertex in \( H \) has degree 1 or 2 . In fact, it is easy to check that every component of \( H \) is an even cycle or an infinite path. Picking every other edge from each component, we obtain a 1-factor of \( G \) . The corresponding path problem in non-bipartite graphs, with sets of disjoint \( A - B \) paths instead of matchings, is less trivial. Let us say that a set \( \mathcal{P} \) of paths in \( G \) covers a set \( U \) of vertices if every vertex in \( U \) is an endvertex of a path in \( \mathcal{P} \) . ## Theorem 8.4.7. (Pym 1969) Let \( G \) be a graph, and let \( A, B \subseteq V\left( G\right) \) . Suppose that \( G \) contains two sets of disjoint \( A - B \) paths, one covering \( A \) and one covering \( B \) . Then \( G \) contains a set of disjoint \( A - B \) paths covering \( A \cup B \) . Some hints for a proof of Theorem 8.4.7 are included with Exercise 52. Next, let us see how the standard matching theorems for finite graphs -König, Hall, Tutte, Gallai-Edmonds -extend to infinite graphs. For locally finite graphs, they all have straightforward extensions by compactness; see Exercises 14-16. But there are also very satisfactory extensions to graphs of arbitrary cardinality. Their proofs form a coherent body of theory and are much deeper, so we shall only be able to state those results and point out how some of them are related. But, as with Menger's theorem, the statements themselves are interesting too: finding the 'right' restatement of a given finite result to make a substantial infinite theorem is by no means easy, and most of them were found only as the theory itself developed over the years. Let us start with bipartite graphs. The following Erdős-Menger-type extension of König's theorem (2.1.1) is now a corollary of Theorem 8.4.2: ## Theorem 8.4.8. (Aharoni 1984) Every bipartite graph has a matching, \( M \) say, and a vertex cover of its edge set that consists of exactly one vertex from every edge in \( M \) . What about an infinite version of the marriage theorem (2.1.2)? The finite theorem says that a matching exists as soon as every subset \( S \) of the first partition class has enough neighbours in the second. But how do we measure 'enough' in an infinite graph? Just as in Menger's theorem, comparing cardinalities is not enough (Exercise 15). However, there is a neat way of rephrasing the marriage condition for a finite graph without appealing to cardinalities. Call a subset \( X \) of one partition class matchable to a subset \( Y \) of the other if the subgraph --- matchable --- spanned by \( X \) and \( Y \) contains a matching of \( X \) . Now if \( S \) is minimal with \( \left| S\right| > \left| {N\left( S\right) }\right| \), then \( S \) is ’larger’ than \( N\left( S\right) \) in the sense that \( S \) is not matchable to \( N\left( S\right) \) but \( N\left( S\right) \) is matchable to \( S \) -by the marriage theorem! (Indeed, by the minimality of \( S \) and the marriage theorem, any \( {S}^{\prime } \subseteq S \) with \( \left| {S}^{\prime }\right| = \left| S\right| - 1 \) can be matched to \( N\left( S\right) \) . As \( \left| {S}^{\prime }\right| = \left| S\right| - 1 \geq \) \( \left. {\left| {N\left( S\right) }\right| \text{, this matching covers}N\left( S\right) \text{.}}\right) \) Thus, if there is any obstruction \( S \) to a perfect matching of the type \( \left| S\right| > \left| {N\left( S\right) }\right| \), there is also one where \( S \) is larger than \( N\left( S\right) \) in this other sense: that \( S \) is not matchable to \( N\left( S\right) \) but \( N\left( S\right) \) is matchable to \( S \) . Rewriting the marriage condition in this way does indeed yield an infinite version of Hall's theorem, which follows from Theorem 8.4.8 just as the marriage theorem follows from König's theorem: Corollary 8.4.9. A bipartite graph with bipartition \( \{ A, B\} \) contains a matching of \( A \) unless there is a set \( S \subseteq A \) such that \( S \) is not matchable to \( N\left( S\right) \) but \( N\left( S\right) \) is matchable to \( S \) . Proof. Consider a matching \( M \) and a cover \( U \) as in Theorem 8.4.8. Then \( U \cap B \supseteq N\left( {A \smallsetminus U}\right) \) is matchable to \( A \smallsetminus U \), by the edges of \( M \) . And if \( A \smallsetminus U \) is matchable to \( N\left( {A \smallsetminus U}\right) \), then adding this matching to the edges of \( M \) incident with

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

where \( S \) is larger than \( N\left( S\right) \) in this other sense: that \( S \) is not matchable to \( N\left( S\right) \) but \( N\left( S\right) \) is matchable to \( S \) . Rewriting the marriage condition in this way does indeed yield an infinite version of Hall's theorem, which follows from Theorem 8.4.8 just as the marriage theorem follows from König's theorem: Corollary 8.4.9. A bipartite graph with bipartition \( \{ A, B\} \) contains a matching of \( A \) unless there is a set \( S \subseteq A \) such that \( S \) is not matchable to \( N\left( S\right) \) but \( N\left( S\right) \) is matchable to \( S \) . Proof. Consider a matching \( M \) and a cover \( U \) as in Theorem 8.4.8. Then \( U \cap B \supseteq N\left( {A \smallsetminus U}\right) \) is matchable to \( A \smallsetminus U \), by the edges of \( M \) . And if \( A \smallsetminus U \) is matchable to \( N\left( {A \smallsetminus U}\right) \), then adding this matching to the edges of \( M \) incident with \( A \cap U \) yields a matching of \( A \) . Applied to a finite graph, Corollary 8.4.9 implies the marriage theorem: if \( N\left( S\right) \) is matchable to \( S \) but not conversely, then clearly \( \left| S\right| > \left| {N\left( S\right) }\right| \) . --- partial matching --- Let us now turn to non-bipartite graphs. If a finite graph has a 1- factor, then the set of vertices covered by any partial matching-one that leaves some vertices unmatched - can be increased by an augmenting path, an alternating path whose first and last vertex are unmatched --- augmenting path --- (Ex. 1, Ch. 2). In an infinite graph we no longer insist that augmenting paths be finite, as long as they have a first vertex. Then, starting at any unmatched vertex with an edge of the 1-factor that we are assuming to exist, we can likewise find a unique maximal alternating path that will either be a ray or end at another unmatched vertex. Switching edges along this path we can then improve our current matching to increase the set of matched vertices, just as in a finite graph. The existence of an inaugmentable partial matching, therefore, is an obvious obstruction to the existence of a 1-factor. The following theorem asserts that this obstruction is the only one: ## Theorem 8.4.10. (Steffens 1977) A countable graph has a 1-factor if and only if for every partial matching there exists an augmenting path. Unlike its finite counterpart, Theorem 8.4.10 is far from trivial: augmenting a given matching 'blindly' need not lead to a well-defined matching at limit steps, since a given edge may get toggled on and off infinitely often (in which case its status will be undefined at the limit example?). We therefore cannot simply find the desired 1-factor inductively. In fact, Theorem 8.4.10 does not extend to uncountable graphs (Exercise 55). However, from the obstruction of inaugmentable partial matchings one can derive a Tutte-type condition that does extend. Given a set \( S \) of vertices in a graph \( G \), let us write \( {\mathcal{C}}_{G - S}^{\prime } \) for the set of \( {\mathcal{C}}_{G - S}^{\prime } \) factor-critical components of \( G - S \), and \( {G}_{S}^{\prime } \) for the bipartite graph with vertex set \( S \cup {\mathcal{C}}_{G - S}^{\prime } \) and edge set \( \{ {sC} \mid \exists c \in C : {sc} \in E\left( G\right) \} \) . Theorem 8.4.11. (Aharoni 1988) \( A \) graph \( G \) has a 1-factor if and only if, for every set \( S \subseteq V\left( G\right) \), the set \( {\mathcal{C}}_{G - S}^{\prime } \) is matchable to \( S \) in \( {G}_{S}^{\prime } \) . Applied to a finite graph, Theorem 8.4.11 implies Tutte's 1-factor theorem (2.2.1): if \( {\mathcal{C}}_{G - S}^{\prime } \) is not matchable to \( S \) in \( {G}_{S}^{\prime } \), then by the marriage theorem there is a subset \( {S}^{\prime } \) of \( S \) that sends edges to more than \( \left| {S}^{\prime }\right| \) components in \( {\mathcal{C}}_{G - S}^{\prime } \) that are also components of \( G - {S}^{\prime } \), and these components are odd because they are factor-critical. Theorems 8.4.8 and 8.4.11 also imply an infinite version of the Gallai-Edmonds theorem (2.2.3): Corollary 8.4.12. Every graph \( G = \left( {V, E}\right) \) has a set \( S \) of vertices that is matchable to \( {\mathcal{C}}_{G - S}^{\prime } \) in \( {G}_{S}^{\prime } \) and such that every component of \( G - S \) not in \( {\mathcal{C}}_{G - S}^{\prime } \) has a 1-factor. Given any such set \( S \), the graph \( G \) has a 1-factor if and only if \( {\mathcal{C}}_{G - S}^{\prime } \) is matchable to \( S \) in \( {G}_{S}^{\prime } \) . Proof. Given a pair \( \left( {S, M}\right) \) where \( S \subseteq V \) and \( M \) is a matching of \( S \) in \( {G}_{S}^{\prime } \), and given another such pair \( \left( {{S}^{\prime },{M}^{\prime }}\right) \), write \( \left( {S, M}\right) \leq \left( {{S}^{\prime },{M}^{\prime }}\right) \) if \[ S \subseteq {S}^{\prime } \subseteq V \smallsetminus \bigcup \left\{ {V\left( C\right) \mid C \in {\mathcal{C}}_{G - S}^{\prime }}\right\} \] and \( M \subseteq {M}^{\prime } \) . Since \( {\mathcal{C}}_{G - S}^{\prime } \subseteq {\mathcal{C}}_{G - {S}^{\prime }}^{\prime } \) for any such \( S \) and \( {S}^{\prime } \), Zorn’s lemma implies that there is a maximal such pair \( \left( {S, M}\right) \) . \( S, M \) For the first statement, we have to show that every component \( C \) of \( G - S \) that is not in \( {\mathcal{C}}_{G - S}^{\prime } \) has a 1-factor. If it does not, then by Theorem 8.4.11 there is a set \( T \subseteq V\left( C\right) \) such that \( {\mathcal{C}}_{C - T}^{\prime } \) is not matchable to \( T \) in \( {C}_{T}^{\prime } \) . By Corollary 8.4.9, this means that \( {\mathcal{C}}_{C - T}^{\prime } \) has a subset \( \mathcal{C} \) that is not matchable in \( {C}_{T}^{\prime } \) to the set \( {T}^{\prime } \subseteq T \) of its neighbours, while \( {T}^{\prime } \) is matchable to \( \mathcal{C} \) ; let \( {M}^{\prime } \) be such a matching. Then \( \left( {S, M}\right) < \) \( \left( {S \cup {T}^{\prime }, M \cup {M}^{\prime }}\right) \), contradicting the maximality of \( \left( {S, M}\right) \) . Of the second statement, only the backward implication is nontrivial. Our assumptions now are that \( {\mathcal{C}}_{G - S}^{\prime } \) is matchable to \( S \) in \( {G}_{S}^{\prime } \) and vice versa (by the choice of \( S \) ), so Proposition 8.4.6 yields that \( {G}_{S}^{\prime } \) has a 1-factor. This defines a matching of \( S \) in \( G \) that picks one vertex \( {x}_{C} \) from every component \( C \in {\mathcal{C}}_{G - S}^{\prime } \) and leaves the other components of \( G - S \) untouched. Adding to this matching a 1-factor of \( C - {x}_{C} \) for every \( C \in {\mathcal{C}}_{G - S}^{\prime } \) and a 1 -factor of every other component of \( G - S \), we obtain the desired 1-factor of \( G \) . Infinite matching theory may seem rather mature and complete as it stands, but there are still fascinating unsolved problems in the Erdős-Menger spirit concerning related discrete structures, such as posets or hypergraphs. We conclude with one about graphs. --- strongly perfect --- Call an infinite graph \( G \) perfect if every induced subgraph \( H \subseteq G \) has a complete subgraph \( K \) of order \( \chi \left( H\right) \), and strongly perfect if \( K \) can always be chosen so that it meets every colour class of some \( \chi \left( H\right) \) - colouring of \( H \) . (Exercise 58 gives an example of a perfect graph that is --- weakly perfect --- not strongly perfect.) Call \( G \) weakly perfect if the chromatic number of every induced subgraph \( H \subseteq G \) is at most the supremum of the orders of its complete subgraphs. ## Conjecture. (Aharoni & Korman 1993) Every weakly perfect graph without infinite independent sets of vertices is strongly perfect. ## 8.5 The topological end space In this last section we shall develop a deeper understanding of the global structure of infinite graphs, especially locally finite ones, that can be attained only by studying their ends. This structure is intrinsically topological, but no more than the most basic concepts of point-set topology will be needed. Our starting point will be to make precise the intuitive idea that the ends of a graph are the 'points at infinity' to which its rays converge. To do so, we shall define a topological space \( \left| G\right| \) associated with a graph \( G = \left( {V, E,\Omega }\right) \) and its ends. \( {}^{8} \) By considering topological versions of paths, cycles and spanning trees in this space, we shall then be able to extend to infinite graphs some parts of finite graph theory that would not otherwise have infinite counterparts (see the notes for more examples). Thus, the ends of an infinite graph turn out to be more than a curious new phenomenon: they form an integral part of the picture, without which it cannot be properly understood. To build the space \( \left| G\right| \) formally, we start with the set \( V \cup \Omega \) . For \( \left( {u, v}\right) \) every edge \( e = {uv} \) we add a set \( \overset{ \circ }{e} = \left( {u, v}\right) \) of continuum many points, making these sets \( \overset{ \circ }{e} \) disjoint from each other and from \( V \cup \Omega \) . We then choose for each \( e \) some fixed bijection between \( \overset{ \circ }{e} \) and the real interval \( \left( {0,1}\right) \), and \( \left\lbrack {u, v}\right\rbrack \) extend this bijection to one between \( \left\lbrack {u, v}\right\rbrack \mathrel{\text{:=}} \{ u\} \cup \mathring{e} \cup \{ v\} \) and \( \left\lbrack {0,1}\right\rbrack \) . This bijection defines a metric on \( \left\lbrack {u, v}\right\rbrack \) ; we call \( \left\lbrack {u, v}\right\rbrack \) a topological edge with inner points \( x \in e \) . Given any \( F \subseteq E \) we write \( F \mathrel{\text{:=}} \bigcup \{ e \mid e \in F\} \) . --- 8 The n

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derstood. To build the space \( \left| G\right| \) formally, we start with the set \( V \cup \Omega \) . For \( \left( {u, v}\right) \) every edge \( e = {uv} \) we add a set \( \overset{ \circ }{e} = \left( {u, v}\right) \) of continuum many points, making these sets \( \overset{ \circ }{e} \) disjoint from each other and from \( V \cup \Omega \) . We then choose for each \( e \) some fixed bijection between \( \overset{ \circ }{e} \) and the real interval \( \left( {0,1}\right) \), and \( \left\lbrack {u, v}\right\rbrack \) extend this bijection to one between \( \left\lbrack {u, v}\right\rbrack \mathrel{\text{:=}} \{ u\} \cup \mathring{e} \cup \{ v\} \) and \( \left\lbrack {0,1}\right\rbrack \) . This bijection defines a metric on \( \left\lbrack {u, v}\right\rbrack \) ; we call \( \left\lbrack {u, v}\right\rbrack \) a topological edge with inner points \( x \in e \) . Given any \( F \subseteq E \) we write \( F \mathrel{\text{:=}} \bigcup \{ e \mid e \in F\} \) . --- 8 The notation of \( \left| G\right| \) comes from topology and clashes with our notation for the order of \( G \) . But there is little danger of confusion, so we keep both. --- When we speak of a ’graph’ \( H \subseteq G \), we shall often also mean its corresponding point set \( V\left( H\right) \cup E\left( H\right) \) . Having thus defined the point set of \( \left| G\right| \), let us choose a basis of open sets to define its topology. For every edge \( {uv} \), declare as open all subsets of \( \left( {u, v}\right) \) that correspond, by our fixed bijection between \( \left( {u, v}\right) \) and \( \left( {0,1}\right) \), to an open set in \( \left( {0,1}\right) \) . For every vertex \( u \) and \( \epsilon > 0 \), declare as open the ’open star around \( u \) of radius \( \epsilon \) ’, that is, the set of all points on edges \( \left\lbrack {u, v}\right\rbrack \) at distance less than \( \epsilon \) from \( u \), measured individually for each edge in its metric inherited from \( \left\lbrack {0,1}\right\rbrack \) . Finally, for every end \( \omega \) and every finite set \( S \subseteq V \), there is a unique component \( C\left( {S,\omega }\right) \) of \( G - S \) that \( C\left( {S,\omega }\right) \) contains a ray from \( \omega \) . Let \( \Omega \left( {S,\omega }\right) \mathrel{\text{:=}} \left\{ {{\omega }^{\prime } \in \Omega \mid C\left( {S,{\omega }^{\prime }}\right) = C\left( {S,\omega }\right) }\right\} \) . For every \( \epsilon > 0 \), write \( {E}_{\epsilon }\left( {S,\omega }\right) \) for the set of all inner points of \( S - \) \( C\left( {S,\omega }\right) \) edges at distance less than \( \epsilon \) from their endpoint in \( C\left( {S,\omega }\right) \) . Then declare as open all sets of the form \[ {\widehat{C}}_{\epsilon }\left( {S,\omega }\right) \mathrel{\text{:=}} C\left( {S,\omega }\right) \cup \Omega \left( {S,\omega }\right) \cup {\overset{ \circ }{E}}_{\epsilon }\left( {S,\omega }\right) . \] \( {\widehat{C}}_{\epsilon }\left( {S,\omega }\right) \) This completes the definition of \( \left| G\right| \), whose open sets are the unions of the sets we explicitly chose as open above. Any subsets of \( \left| G\right| \) we consider, including those that are ’graphs’ in their own right (i.e., unions of vertices and topological edges of \( G \) ), will carry the subspace topology in \( \left| G\right| \) . Such sets may contain ends of \( G \) , quite independently of whether they contain any rays from such ends: --- standard subspace --- they are just subsets of the point set \( \left| G\right| \cdot {}^{9} \) A standard subspace of \( \left| G\right| \) is one that contains every edge (including its endvertices) of which it contains an inner point. The closure of a set \( X \subseteq \left| G\right| \) will be denoted by \( \bar{X} \) . For example, closure \( \bar{X} \) \( \bar{V} = V \cup \Omega \) (because every neighbourhood of an end contains a vertex), and the closure of a ray is obtained by adding its end. More generally, if \( X \subseteq V \) is the set of teeth of a comb then \( \bar{X} \) contains the end of its spine, while conversely for every end \( \omega \in \bar{X} \) and every ray \( R \in \omega \) there is a comb with spine \( R \) and teeth in \( X \) (Exercise 59). In particular, the closure of the subgraph \( C\left( {S,\omega }\right) \) considered above is the set \( C\left( {S,\omega }\right) \cup \Omega \left( {S,\omega }\right) \) . By definition, \( \left| G\right| \) is always Hausdorff. When \( G \) is connected and locally finite, then \( \left| G\right| \) is also compact: \( {}^{10} \) Proposition 8.5.1. If \( G \) is connected and locally finite, then \( \left| G\right| \) is a compact Hausdorff space. Proof. Let \( \mathcal{O} \) be an open cover of \( \left| G\right| \) ; we show that \( \mathcal{O} \) has a finite (8.1.2) subcover. Pick a vertex \( {v}_{0} \in G \), write \( {D}_{n} \) for the (finite) set of vertices at distance \( n \) from \( {v}_{0} \), and put \( {S}_{n} \mathrel{\text{:=}} {D}_{0} \cup \ldots \cup {D}_{n - 1} \) . For every \( v \in {D}_{n} \) , let \( C\left( v\right) \) denote the component of \( G - {S}_{n} \) containing \( v \), and let \( \widehat{C}\left( v\right) \) be --- 9 Except in Exercise 62, we never consider the ends of subgraphs as such. 10 Topologists call \( \left| G\right| \) the Freudenthal compactification of \( G \) . --- its closure together with all inner points of \( C\left( v\right) - {S}_{n} \) edges. Then \( G\left\lbrack {S}_{n}\right\rbrack \) and these \( \widehat{C}\left( v\right) \) together partition \( \left| G\right| \) . We wish to prove that, for some \( n \), each of the sets \( \widehat{C}\left( v\right) \) with \( v \in {D}_{n} \) is contained in some \( O\left( v\right) \in \mathcal{O} \) . For then we can take a finite subcover of \( \mathcal{O} \) for \( G\left\lbrack {S}_{n}\right\rbrack \) (which is compact, being a finite union of edges and vertices), and add to it these finitely many sets \( O\left( v\right) \) to obtain the desired finite subcover for \( \left| G\right| \) . Suppose there is no such \( n \) . Then for each \( n \) the set \( {V}_{n} \) of vertices \( v \in {D}_{n} \) such that no set from \( \mathcal{O} \) contains \( \widehat{C}\left( v\right) \) is non-empty. Moreover, for every neighbour \( u \in {D}_{n - 1} \) of \( v \in {V}_{n} \) we have \( C\left( v\right) \subseteq C\left( u\right) \) because \( {S}_{n - 1} \subseteq {S}_{n} \), and hence \( u \in {V}_{n - 1} \) ; let \( f\left( v\right) \) be such a vertex \( u \) . By the infinity lemma (8.1.2) there is a ray \( R = {v}_{0}{v}_{1}\ldots \) with \( {v}_{n} \in {V}_{n} \) for all \( n \) . Let \( \omega \) be its end, and let \( O \in \mathcal{O} \) contain \( \omega \) . Since \( O \) is open, it contains a basic open neighbourhood of \( \omega \) : there exist a finite set \( S \subseteq V \) and \( \epsilon > 0 \) such that \( {\widehat{C}}_{\epsilon }\left( {S,\omega }\right) \subseteq O \) . Now choose \( n \) large enough that \( {S}_{n} \) contains \( S \) and all its neighbours. Then \( \widehat{C}\left( {v}_{n}\right) \) lies inside a component of \( G - S \) . As \( C\left( {v}_{n}\right) \) contains \( {v}_{n}R \in \omega \), this component must be \( C\left( {S,\omega }\right) \) . Thus \[ \widehat{C}\left( {v}_{n}\right) \subseteq C\left( {S,\omega }\right) \subseteq O \in \mathcal{O}, \] contradicting the fact that \( {v}_{n} \in {V}_{n} \) . If \( G \) has a vertex of infinite degree then \( \left| G\right| \) cannot be compact. (Why not?) But \( \Omega \left( G\right) \) can be compact; see Exercise 61 for when it is. What else can we say about the space \( \left| G\right| \) in general? For example, is it metrizable? Using a normal spanning tree \( T \) of \( G \), it is indeed not difficult to define a metric on \( \left| G\right| \) that induces its topology. But not every connected graph has a normal spanning tree, and it is not easy to determine which graphs do. Surprisingly, though, it is possible conversely to deduce the existence of a normal spanning tree just from the assumption that the subspace \( V \cup \Omega \) of \( \left| G\right| \) is metric. Thus whenever \( \left| G\right| \) is metrizable, a natural metric can be made visible in this simple structural way: Theorem 8.5.2. For a connected graph \( G \), the space \( \left| G\right| \) is metrizable if and only if \( G \) has a normal spanning tree. The proof of Theorem 8.5.2 is indicated in Exercises 30 and 63. Our next aim is to review, or newly define, some topological notions of paths and connectedness, of cycles, and of spanning trees. By substituting these topological notions with respect to \( \left| G\right| \) for the corresponding graph-theoretical notions with respect to \( G \), one can extend to locally finite graphs a number of theorems about paths, cycles and spanning trees in finite graphs that would not otherwise extend. We shall do this, as a case in point, for the tree-packing theorem of Nash-Williams and Tutte (Theorem 2.4.1); references for more such results are given in the notes. Let \( X \) be an arbitrary Hausdorff space. (Later, this will be a subspace of \( \left| G\right| \) .) \( X \) is (topologically) connected if it is not a union of two connected disjoint non-empty open subsets. If we think of two points of \( X \) as equivalent if \( X \) has a connected subspace containing both, we have an equivalence relation whose classes are the (connected) components of \( X \) . component These are the maximal connected subspaces of \( X \) . Components are always closed, but if \( X \) has infinitely many components they need not be open. We shall need the following lemma; see the notes for a reference. Lemma 8.5.3. If \( X \) is compact and \( {A}_{1},{A}_{2} \) are distinct components of \( X \), then \( X \) is a union of disjoint open sets \( {X}_{1},{X}_{2} \) suc

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ot otherwise extend. We shall do this, as a case in point, for the tree-packing theorem of Nash-Williams and Tutte (Theorem 2.4.1); references for more such results are given in the notes. Let \( X \) be an arbitrary Hausdorff space. (Later, this will be a subspace of \( \left| G\right| \) .) \( X \) is (topologically) connected if it is not a union of two connected disjoint non-empty open subsets. If we think of two points of \( X \) as equivalent if \( X \) has a connected subspace containing both, we have an equivalence relation whose classes are the (connected) components of \( X \) . component These are the maximal connected subspaces of \( X \) . Components are always closed, but if \( X \) has infinitely many components they need not be open. We shall need the following lemma; see the notes for a reference. Lemma 8.5.3. If \( X \) is compact and \( {A}_{1},{A}_{2} \) are distinct components of \( X \), then \( X \) is a union of disjoint open sets \( {X}_{1},{X}_{2} \) such that \( {A}_{1} \subseteq {X}_{1} \) and \( {A}_{2} \subseteq {X}_{2} \) . An \( \operatorname{arc} \) in \( X \) is a homeomorphic image in \( X \) of the real unit interval \( \left\lbrack {0,1}\right\rbrack \) ; it links the images of 0 and 1, which are its endpoints. Being linked by an arc is also an equivalence relation on \( X \) (since every \( x - y \) component arc \( A \) has a first point \( p \) on any \( y - z \) arc \( B \), because \( B \) is closed, so \( {ApB} \) is an \( x - z \) arc); the equivalence classes are the arc-components of \( X \) . If there is only one arc-component, then \( X \) is arc-connected. Since \( \left\lbrack {0,1}\right\rbrack \) connected is connected, arc-connectedness implies connectedness. The converse implication is false in general, even for spaces \( X \subseteq \left| G\right| \) with \( G \) locally finite. But it holds in an important special case: Lemma 8.5.4. If \( G \) is a locally finite graph, then every closed connected subspace of \( \left| G\right| \) is arc-connected. The proof of Lemma 8.5.4 is not easy; see the notes for a reference. Every finite path in \( G \) defines an arc in \( \left| G\right| \) in an obvious way. Similarly, every ray is an arc linking its starting vertex to its end, and a double ray in \( G \) forms an arc in \( \left| G\right| \) together with the two ends of its tails, if these ends are distinct. Consider an end \( \omega \) in a standard subspace \( X \) of \( \left| G\right| \), and \( k \in \mathbb{N} \cup \{ \infty \} \) . If \( k \) is the maximum number of arcs in \( X \) that have \( \omega \) as their common endpoint and are otherwise disjoint, then \( k \) is the (topological) vertex-degree of \( \omega \) in \( X \) . The (topological) edge-degree --- end degrees in subspaces --- of \( \omega \) in \( X \) is defined analogously, using edge-disjoint arcs. In analogy to Theorem 8.2.5 one can show that these maxima are always attained, so every end of \( G \) that lies in \( X \) has a topological vertex- and edge-degree there. For \( X = \left| G\right| \) and \( G \) locally finite, the (topological) end degrees in \( X \) coincide with the combinatorial end degrees defined earlier. Unlike finite paths, arcs can jump across a vertex partition without containing an edge from the corresponding cut, provided the cut is infinite: Lemma 8.5.5. Let \( G \) be connected and locally finite, \( \{ X, Y\} \) a partition of \( V\left( G\right) \), and \( F \mathrel{\text{:=}} E\left( {X, Y}\right) \) . (i) \( F \) is finite if and only if \( \bar{X} \cap \bar{Y} = \varnothing \) . (ii) If \( F \) is finite, there is no arc in \( \left| G\right| \smallsetminus \overset{ \circ }{F} \) with one endpoint in \( X \) and the other in \( Y \) . (iii) If \( F \) is infinite and \( X \) and \( Y \) are both connected in \( G \), there is such an arc. (8.2.2) Proof. (i) Suppose first that \( F \) is infinite. Since \( G \) is locally finite, the set \( {X}^{\prime } \) of endvertices of \( F \) in \( X \) is also infinite. By the star-comb lemma (8.2.2), there is a comb in \( G \) with teeth in \( {X}^{\prime } \) ; let \( \omega \) be the end of its spine. Then every basic open neighbourhood \( {\widehat{C}}_{\epsilon }\left( {S,\omega }\right) \) of \( \omega \) meets \( {X}^{\prime } \) infinitely and hence also meets \( Y \), giving \( \omega \in \bar{X} \cap \bar{Y} \) . Suppose now that \( F \) is finite. Let \( S \) be the set of vertices incident with edges in \( F \) . Then \( S \) is finite and separates \( X \) from \( Y \) . Since every basic open set of the form \( {\widehat{C}}_{\epsilon }\left( {S,\omega }\right) \) misses \( X \) or \( Y \), no end \( \omega \) lies in the closure of both. (ii) Clearly \( \left| G\right| \smallsetminus \overset{ \circ }{F} = \overline{G\left\lbrack X\right\rbrack } \cup \overline{G\left\lbrack Y\right\rbrack } \), and by (i) this union is disjoint. Hence no connected subset of \( \left| G\right| \smallsetminus F \) can meet both \( X \) and \( Y \) , but arcs are continuous images of \( \left\lbrack {0,1}\right\rbrack \) and hence connected. (iii) By (i), there is an end \( \omega \in \bar{X} \cap \bar{Y} \) . Apply the star-comb lemma in \( G\left\lbrack X\right\rbrack \) to any sequence of vertices in \( X \) converging to \( \omega \) ; this yields a comb whose spine \( R \) lies in \( \omega \) . Similarly, there is a comb in \( G\left\lbrack Y\right\rbrack \) whose spine \( {R}^{\prime } \) lies in \( \omega \) . Now \( R \cup \{ \omega \} \cup {R}^{\prime } \) is the desired arc. circle A circle in a topological space is a homeomorphic image of the unit circle \( {S}^{1} \subseteq {\mathbb{R}}^{2} \) . For example, if \( G \) is the 2-way infinite ladder shown in Figure 8.1.3, and we delete all its rungs (the vertical edges), what remains is a disjoint union \( D \) of two double rays; the closure of \( D \) in \( \left| G\right| \) , obtained by adding the two ends of \( G \), is a circle. Similarly, the double ray 'around the outside' of the 1-way ladder forms a circle together with the unique end of that ladder. A more adventurous example of a circle is shown in Figure 8.5.1. Let \( G \) be the graph obtained from the binary tree \( {T}_{2} \) by joining for every finite \( 0 - 1 \) sequence \( \ell \) the vertices \( \ell {01} \) and \( \ell {10} \) by a new edge \( {e}_{\ell } \) . Together with all the ends of \( G \), the double rays \( {D}_{\ell } \ni {e}_{\ell } \) shown in the figure form an arc \( A \) in \( \left| G\right| \), whose union with the bottom double ray \( D \) is a circle in \( \left| G\right| \) (Exercise 69). Note that no two of the double rays in \( A \) are consecutive: between any two there lies a third. This is why end degrees in subspaces are defined in terms of arcs rather than rays, so that the ends in a circle can always have degree 2 in it. And indeed they do (Exercise 70):  Fig. 8.5.1. A circle containing uncountably many ends Lemma 8.5.6. Let \( G \) be locally finite. A closed standard subspace \( C \) of \( \left| G\right| \) is a circle in \( \left| G\right| \) if and only if \( C \) is connected, every vertex in \( C \) is incident with exactly two edges in \( C \), and every end in \( C \) has vertex-degree 2 (equivalently: edge-degree 2) in \( C \) . It is not hard to show that every circle \( C \) in a space \( \left| G\right| \) is a standard subspace; the set \( D \) of edges it contains will be called its circuit. Then circuit \( C \) is the closure of the point set \( \bigcup D \), as every neighbourhood in \( C \) of a vertex or end meets an edge, which must then be contained in \( C \) and hence lie in \( D \) . In particular, there are no circles consisting only of ends, and every circle is uniquely determined by its circuit. --- topological spanning tree --- A topological spanning tree of \( G \) is an arc-connected standard subspace of \( \left| G\right| \) that contains every vertex and every end but contains no circle. Clearly, such a subspace \( X \) must be closed. With respect to the addition or deletion of edges, it is both minimally arc-connected and maximally 'acirclic'. As with ordinary trees, one can show that every two points of \( X \) are joined by a unique arc in \( X \) . Thus, adding a new edge \( e \) to \( X \) creates a unique circle in \( X \cup e \) ; its edges form the fundamental circuit \( {C}_{e} \) of \( e \) with respect to \( X \) . Similarly, for every edge \( e \subseteq X \) the space \( X \smallsetminus \overset{ \circ }{e} \) has exactly two arc-components; the set of edges between these is the fundamental cut \( {D}_{e} \) . If \( G \) is locally finite, then its fundamental cuts are finite (Exercise 74). One might expect that the closure \( \bar{T} \) of an ordinary spanning tree \( T \) of \( G \) is always a topological spanning tree of \( \left| G\right| \) . However, this can fail in two ways: if \( T \) has a vertex of infinite degree then \( \bar{T} \) may fail to be arc-connected (although it will be topologically connected, because \( T \) is); if \( T \) is locally finite, then \( \bar{T} \) will be arc-connected but may contain a circle (Figure 8.5.2). On the other hand, a subgraph whose closure is a topological spanning tree may well be disconnected: the vertical rays  Fig. 8.5.2. \( {\bar{T}}_{1} \) is a topological spanning tree, but \( {\bar{T}}_{2} \) contains three circles in the \( \mathbb{N} \times \mathbb{N} \) grid, for example, form a topological spanning tree of the grid together with its unique end. In general, there seems to be no canonical way to construct topological spanning trees, and it is unknown whether every connected graph has one. Countab

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) is always a topological spanning tree of \( \left| G\right| \) . However, this can fail in two ways: if \( T \) has a vertex of infinite degree then \( \bar{T} \) may fail to be arc-connected (although it will be topologically connected, because \( T \) is); if \( T \) is locally finite, then \( \bar{T} \) will be arc-connected but may contain a circle (Figure 8.5.2). On the other hand, a subgraph whose closure is a topological spanning tree may well be disconnected: the vertical rays  Fig. 8.5.2. \( {\bar{T}}_{1} \) is a topological spanning tree, but \( {\bar{T}}_{2} \) contains three circles in the \( \mathbb{N} \times \mathbb{N} \) grid, for example, form a topological spanning tree of the grid together with its unique end. In general, there seems to be no canonical way to construct topological spanning trees, and it is unknown whether every connected graph has one. Countable connected graphs, however, do have topolo- (8.2.4) gical spanning trees, by Theorem 8.2.4: ## Lemma 8.5.7. The closure of any normal spanning tree is a topological spanning tree. (1.5.5) Proof. Let \( T \) be a normal spanning tree of \( G \) . By Lemma 8.2.3, every \( \left( {8.2.3}\right) \) end \( \omega \) of \( G \) contains a normal ray \( R \) of \( T \) . Then \( R \cup \{ \omega \} \) is an arc linking \( \omega \) to the root of \( T \), so \( \bar{T} \) is arc-connected. It remains to check that \( \bar{T} \) contains no circle. Suppose it does, and let \( A \) be the \( u - v \) arc obtained from that circle by deleting the inner points of an edge \( e = {uv} \) it contains. Clearly, \( e \in T \) . Assume that \( u < v \) in the tree-order of \( T \), let \( {T}_{u} \) and \( {T}_{v} \) denote the components of \( T - e \) containing \( u \) and \( v \), and notice that \( V\left( {T}_{v}\right) \) is the up-closure \( \lfloor v\rfloor \) of \( v \) in \( T \) . Now let \( S \mathrel{\text{:=}} \lceil u\rceil \) . By Lemma 1.5.5(ii), \( \lfloor v\rfloor \) is the vertex set of a component \( C \) of \( G - S \) . Thus, \( V\left( C\right) = V\left( {T}_{v}\right) \) and \( V\left( {G - C}\right) = V\left( {T}_{u}\right) \) , so the set \( E\left( {C, S}\right) \) of edges between these sets contains no edge of \( A \) . But \( \bar{C} \) and \( \widehat{G - C} \) partition \( \left| G\right| \smallsetminus \mathring{E}\left( {C, S}\right) \) into two open sets. \( {}^{11} \) As \( A \subseteq \left| G\right| \smallsetminus E\left( {C, S}\right) \), this contradicts the fact that \( A \) is topologically connected. We now extend the notion of the cycle space to locally finite infinite graphs \( G \), based on their (possibly infinite) circuits. thin Call a family \( {\left( {D}_{i}\right) }_{i \in I} \) of subsets of \( E\left( G\right) \) thin if no edge lies in \( {D}_{i} \) sum for infinitely many \( i \) . Let the sum \( \mathop{\sum }\limits_{{i \in I}}{D}_{i} \) of this family be the set --- topological cycle space --- of all edges that lie in \( {D}_{i} \) for an odd number of indices \( i \) . Now define the (topological) cycle space \( \mathcal{C}\left( G\right) \) of \( G \) as the subspace of its edge space \( \mathcal{E}\left( G\right) \) consisting of all sums of (thin families of) circuits. (Note that \( \mathcal{C}\left( G\right) \) is closed under addition: just combine the two thin families into one.) Clearly, this definition of \( \mathcal{C}\left( G\right) \) agrees with that from Chapter 1.9 when \( G \) is finite. --- 11 Open in the subspace topology: add \( \mathring{E}\left( {C, S}\right) \) to obtain open sets in \( \left| G\right| \) . --- We say that a given set \( \mathcal{Z} \) of circuits generates \( \mathcal{C}\left( G\right) \) if every element generates of \( \mathcal{C}\left( G\right) \) is a sum of elements of \( \mathcal{Z} \) . For example, the cycle space of the ladder in Figure 8.1.3 can be generated by all its squares (the 4-element circuits), or by the infinite circuit consisting of all horizontal edges and all squares but one. Similarly, the 'wild' circuit of Figure 8.5.1 is the sum of all the finite face boundaries in that graph. The following two theorems summarize how the properties of the cycle spaces of finite graphs, familiar from Chapter 1, extend to locally finite graphs with topological cycle spaces. Theorem 8.5.8. (Diestel & Kühn 2004) Let \( G = \left( {V, E,\Omega }\right) \) be a locally finite connected graph. (i) \( \mathcal{C}\left( G\right) \) contains precisely those subsets of \( E \) that meet every finite cut in an even number of edges. (ii) Every element of \( \mathcal{C}\left( G\right) \) is a disjoint sum of circuits. (iii) The fundamental circuits of any topological spanning tree of \( G \) generate \( \mathcal{C}\left( G\right) \) . While the proofs of parts (i) and (iii) of Theorem 8.5.8 are straightforward, part (ii) is not that easy. This is because it is no longer straightforward to isolate a single circuit from a given element of \( \mathcal{C}\left( G\right) \) . For example, we know that the 'wild' circuit of the graph in Figure 8.5.1 must lie in its cycle space, since it is clearly the sum of the finite circuits bounding a face. But in order to construct a 'decomposition' of this element of \( \mathcal{C}\left( G\right) \) into ’disjoint circuits’, the proof of (ii) has to, somehow, construct this circuit without appealing to the special structure of the graph. Our proof below circumvents these difficulties by appealing to our unproved Lemma 8.5.4 that closed connected subsets of \( \left| G\right| \) are arc-connected, and to the unproved topological Lemma 8.5.3. Proof of Theorem 8.5.8. (i) Let \( D \in \mathcal{C}\left( G\right) \) be given, and consider --- (1.5.5) (8.2.4) --- a finite cut \( F \) . By definition, \( D \) is a sum of a thin family of circuits. Only finitely many of these can meet \( F \), so it suffices to show that every circuit meets \( F \) evenly. To prove this, consider a circle \( C \) in \( \left| G\right| \) . As \( F \) is a finite cut, any arc in \( \left| G\right| \) that links the two sides of the corresponding vertex partition contains an edge from \( F \), by Lemma 8.5.5 (ii). Hence every arc on \( C \) between two consecutive edges from \( F \) links these at their endvertices on the same side of \( F \), which implies that \( C \) contains an even number of edges from \( F \) . Conversely, let \( D \) be any set of edges that meets every finite cut evenly. Let \( T \) be a normal spanning tree of \( G \) (Theorem 8.2.4). We claim that \[ D = \mathop{\sum }\limits_{{e \in D \smallsetminus E\left( T\right) }}{C}_{e} \] \( \left( *\right) \) where \( {C}_{e} \) denotes the fundamental circuit of \( e \) with respect to \( T \) . To prove this, consider the edges \( f \) of \( G \) separately. If \( f \notin T \), then clearly \( f \in D \) if and only if \( f \) lies in the sum in \( \left( *\right) \), since \( {C}_{f} \) is the unique fundamental circuit containing \( f \) . Suppose now that \( f \in T \) . Then \( f \) lies in precisely those \( {C}_{e} \) for which \( e \) lies in the fundamental cut \( {D}_{f} \) of \( f \) . Thus all we need to show is that \( {D}_{f} \) is finite: then \( D \cap {D}_{f} \) is even by assumption, so \( f \in D \) if and only if an odd number of other edges \( e \in {D}_{f} \) lie in \( D \), which is the case if and only if \( f \) lies in the sum in \( \left( *\right) \) . (In particular, the sum is one over a thin family, and hence well-defined.) To show that \( {D}_{f} \) is finite, assume that \( f = {xy} \) with \( x < y \) in the tree-order of \( T \) . Then the up-closure \( \lfloor y\rfloor \) of \( y \) in \( T \) is one of the two components of \( T - f \), and by Lemma 1.5.5 it spans a component of \( G - \lceil x\rceil \) . Hence every edge in \( {D}_{f} \) has one endvertex in \( \lfloor y\rfloor \) and the other in \( \lceil x\rceil \) . As \( \lceil x\rceil \) is finite and \( G \) is locally finite, this means that there are only finitely many such edges. (ii) Let \( D \in \mathcal{C}\left( G\right) \) be given. Consider a maximal set of disjoint circuits contained in \( D \), and let \( Z \) be their union. Clearly \( Z \in \mathcal{C}\left( G\right) \), and hence \( {Z}^{\prime } \mathrel{\text{:=}} D - Z \in \mathcal{C}\left( G\right) \) . We wish to show that \( {Z}^{\prime } = \varnothing \) . Suppose not. Let \( e = {uv} \) be an edge in \( {Z}^{\prime } \) and put \[ X \mathrel{\text{:=}} \left( {V \cup \Omega \cup \bigcup {Z}^{\prime }}\right) \smallsetminus \overset{ \circ }{e}. \] Clearly, \( X \) is a closed in \( \left| G\right| \), and hence is a compact subspace (Proposition 8.5.1). Let us show that \( u \) and \( v \) lie in different components of \( X \) . If they lie in the same component, \( A \) say, then \( A \) is closed in \( X \) (being a component) and hence in \( \left| G\right| \), so \( A \) is arc-connected by Lemma 8.5.4. But any \( u - v \) arc in \( A \) forms a circle with \( e \) that contradicts the maximality of \( Z \) . Thus, \( u \) and \( v \) lie in different components of \( X \) . By Lemma 8.5.3, \( X \) is a union of disjoint open subsets \( {X}_{u} \ni u \) and \( {X}_{v} \ni v \) . Put \( {V}_{u} \mathrel{\text{:=}} {X}_{u} \cap V \) and \( {V}_{v} \mathrel{\text{:=}} {X}_{v} \cap V \) . As \( {X}_{u} \) and \( {X}_{v} \) are complements in \( X \), they are closed (as well as open) in \( X \) and hence closed in \( \left| G\right| \), so \( \overline{{V}_{u}} \subseteq {X}_{u} \) and \( \overline{{V}_{v}} \subseteq {X}_{v} \) . In particular, \( \overline{{V}_{u}} \cap \overline{{V}_{v}} = \varnothing \) , so by Lemma 8.5.5 (i) the cut \( F \mathrel{\text{:=}} E\left( {{V}_{u},{V}_{v}}\right) \) of \( G \) is finite. Moreover, \( F \cap {Z}^{\prime } = \{ e\} \),

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

\) is closed in \( X \) (being a component) and hence in \( \left| G\right| \), so \( A \) is arc-connected by Lemma 8.5.4. But any \( u - v \) arc in \( A \) forms a circle with \( e \) that contradicts the maximality of \( Z \) . Thus, \( u \) and \( v \) lie in different components of \( X \) . By Lemma 8.5.3, \( X \) is a union of disjoint open subsets \( {X}_{u} \ni u \) and \( {X}_{v} \ni v \) . Put \( {V}_{u} \mathrel{\text{:=}} {X}_{u} \cap V \) and \( {V}_{v} \mathrel{\text{:=}} {X}_{v} \cap V \) . As \( {X}_{u} \) and \( {X}_{v} \) are complements in \( X \), they are closed (as well as open) in \( X \) and hence closed in \( \left| G\right| \), so \( \overline{{V}_{u}} \subseteq {X}_{u} \) and \( \overline{{V}_{v}} \subseteq {X}_{v} \) . In particular, \( \overline{{V}_{u}} \cap \overline{{V}_{v}} = \varnothing \) , so by Lemma 8.5.5 (i) the cut \( F \mathrel{\text{:=}} E\left( {{V}_{u},{V}_{v}}\right) \) of \( G \) is finite. Moreover, \( F \cap {Z}^{\prime } = \{ e\} \), since every other edge of \( {Z}^{\prime } \) lies in \( X \), and hence in \( {X}_{u} \) or in \( {X}_{v} \) . As \( {Z}^{\prime } \in \mathcal{C}\left( G\right) \), this contradicts (i). (iii) In our proof of (i) we already proved the most important case of (iii), where the topological spanning tree in question is the closure of a normal spanning tree. The proof for arbitrary topological spanning trees is the same, except for the proof that all their fundamental cuts are finite (Exercise 74). Corollary 8.5.9. \( \mathcal{C}\left( G\right) \) is generated by finite circuits, and is closed under infinite (thin) sums. Proof. By Theorem 8.2.4, \( G \) has a normal spanning tree, \( T \) say. By (8.2.4) Lemma 8.5.7, its closure \( \bar{T} \) in \( \left| G\right| \) is a topological spanning tree. The fundamental circuits of \( \bar{T} \) coincide with those of \( T \), and are therefore finite. By Theorem 8.5.8 (iii), they generate \( \mathcal{C}\left( G\right) \) . Let \( \mathop{\sum }\limits_{{i \in I}}{D}_{i} \) be a sum of elements of \( \mathcal{C}\left( G\right) \) . By Theorem 8.5.8 (ii), each \( {D}_{i} \) is a disjoint union of circuits. Together, these form a thin family, whose sum equals \( \mathop{\sum }\limits_{{i \in I}}{D}_{i} \) and lies in \( \mathcal{C}\left( G\right) \) . To complete this section, we apply our new notions to extend the tree-packing theorem of Nash-Williams and Tutte (2.4.1) to locally finite graphs. Note that all our definitions extend naturally to multigraphs. Theorem 8.5.10. The following statements are equivalent for all \( k \in \mathbb{N} \) \( k \) and locally finite multigraphs \( G \) : \( G \) (i) \( G \) has \( k \) edge-disjoint topological spanning trees. (ii) For every finite partition of \( V\left( G\right) \), into \( \ell \) sets say, \( G \) has at least \( k\left( {\ell - 1}\right) \) cross-edges. We begin our proof of Theorem 8.5.10 with a compactness extension of the finite theorem, which will give us a slightly weaker statement at the limit. Following Tutte, let us call a spanning submultigraph \( H \) of \( G \) --- semi-connected --- semiconnected in \( G \) if every finite cut of \( G \) contains an edge of \( H \) . Lemma 8.5.11. If for every finite partition of \( V\left( G\right) \), into \( \ell \) sets say, \( G \) has at least \( k\left( {\ell - 1}\right) \) cross-edges, then \( G \) has \( k \) edge-disjoint semicon-nected spanning subgraphs. Proof. Pick an enumeration \( {v}_{0},{v}_{1},\ldots \) of \( V\left( G\right) \) . For every \( n \in \mathbb{N} \) let \( {G}_{n} \) (8.1.2) be the finite multigraph obtained from \( G \) by contracting every component of \( G - \left\{ {{v}_{0},\ldots ,{v}_{n}}\right\} \) to a vertex, deleting any loops but no parallel edges that arise in the contraction. Then \( G\left\lbrack {{v}_{0},\ldots ,{v}_{n}}\right\rbrack \) is an induced submultigraph of \( {G}_{n} \) . Let \( {\mathcal{V}}_{n} \) denote the set of all \( k \) -tuples \( \left( {{H}_{n}^{1},\ldots ,{H}_{n}^{k}}\right) \) of edge-disjoint connected spanning subgraphs of \( {G}_{n} \) . Since every partition \( P \) of \( V\left( {G}_{n}\right) \) induces a partition of \( V\left( G\right) \), since \( G \) has enough cross-edges for that partition, and since all these cross-edges are also cross-edges of \( P \), Theorem 2.4.1 implies that \( {\mathcal{V}}_{n} \neq \varnothing \) . Since every \( \left( {{H}_{n}^{1},\ldots ,{H}_{n}^{k}}\right) \in {\mathcal{V}}_{n} \) induces an element \( \left( {{H}_{n - 1}^{1},\ldots ,{H}_{n - 1}^{k}}\right) \) of \( {\mathcal{V}}_{n - 1} \), the infinity lemma (8.1.2), yields a sequence \( {\left( {H}_{n}^{1},\ldots ,{H}_{n}^{k}\right) }_{n \in \mathbb{N}} \) of \( k \) -tuples, one from each \( {\mathcal{V}}_{n} \), with a limit \( \left( {{H}^{1},\ldots ,{H}^{k}}\right) \) defined by the nested unions \[ {H}^{i} \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{n \in \mathbb{N}}}{H}_{n}^{i}\left\lbrack {{v}_{0},\ldots ,{v}_{n}}\right\rbrack \] These \( {H}^{i} \) are edge-disjoint for distinct \( i \) (because the \( {H}_{n}^{i} \) are), but they need not be connected. To show that they are semiconnected in \( G \) , consider a finite cut \( F \) of \( G \) . Choose \( n \) large enough that all the end-vertices of edges in \( F \) are among \( {v}_{0},\ldots ,{v}_{n} \) . Then \( F \) is also a cut of \( {G}_{n} \) . Now consider the \( k \) -tuple \( \left( {{H}_{n}^{1},\ldots ,{H}_{n}^{k}}\right) \) which the infinity lemma picked from \( {\mathcal{V}}_{n} \) . Each of these \( {H}_{n}^{i} \) is a connected spanning subgraph of \( {G}_{n} \), so it contains an edge from \( F \) . But \( {H}_{n}^{i} \) agrees with \( {H}^{i} \) on \( \left\{ {{v}_{0},\ldots ,{v}_{n}}\right\} \), so \( {H}^{i} \) too contains this edge from \( F \) . At first glance, the notion of semiconnectedness appears to be somewhat ad-hoc: it summarizes what happens to be left of the connectedness of the graphs \( {H}_{n}^{i} \) at their limit \( {H}^{i} \) -and this, no doubt, is why Tutte introduced it. In our context, however, it acquires an unexpected natural meaning: Lemma 8.5.12. A spanning subgraph \( H \subseteq G \) is semiconnected in \( G \) if and only if its closure \( \bar{H} \) in \( \left| G\right| \) is topologically connected. Proof. If \( \bar{H} \) is disconnected, it is contained in the union of two closed subsets \( {O}_{1},{O}_{2} \) of \( \left| G\right| \) that both meet \( \bar{H} \) and satisfy \( {O}_{1} \cap {O}_{2} \cap \bar{H} = \varnothing \) . Since \( \bar{H} \) is a standard subspace containing \( V\left( G\right) \), the sets \( {O}_{i} \) partition \( V\left( G\right) \) into two non-empty sets \( {X}_{1},{X}_{2} \) . Then \[ {\bar{X}}_{1} \cap {\bar{X}}_{2} \subseteq {O}_{1} \cap {O}_{2} \cap \Omega \left( G\right) \subseteq {O}_{1} \cap {O}_{2} \cap \bar{H} = \varnothing . \] By Lemma 8.5.5 (i), this implies that \( G \) has only finitely many \( {X}_{1} - {X}_{2} \) edges. As edges are connected, none of them can lie in \( H \) . Hence, \( H \) is not semiconnected. The converse implication is straightforward (and not needed in our proof of Theorem 8.5.10): a finite cut of \( G \) containing no edge of \( H \) defines a partition of \( \bar{H} \) into non-empty open subsets, showing that \( \bar{H} \) is disconnected. Lemma 8.5.13. Every closed, connected, standard subspace \( X \) of \( \left| G\right| \) that contains \( V\left( G\right) \) also contains a topological spanning tree of \( G \) . Proof. By Lemma 8.5.4, \( X \) is arc-connected. Since \( X \) contains all vertices, \( G \) cannot be disconnected, so its local finiteness implies that it is countable. Let \( {e}_{0},{e}_{1},\ldots \) be an enumeration of the edges in \( X \) . We now delete these edges one by one, keeping \( X \) arc-connected. Starting with \( {X}_{0} \mathrel{\text{:=}} X \), we define \( {X}_{n + 1} \mathrel{\text{:=}} {X}_{n} \smallsetminus {e}_{n} \) if this keeps \( {X}_{n + 1} \) arc-connected; if not, we put \( {X}_{n + 1} \mathrel{\text{:=}} {X}_{n} \) . Finally, we put \( T \mathrel{\text{:=}} \mathop{\bigcap }\limits_{{n \in \mathbb{N}}}{X}_{n} \) . Clearly, \( T \) is closed, contains every vertex and every end of \( G \), but contains no circle: any circle in \( T \) would contain an edge, which should have got deleted. To show that \( T \) is arc-connected, it suffices by Lemmas 8.5.4 and 8.5.12 to check that every finite cut of \( G \) contains an edge from \( T \) . By Lemma 8.5.5 (ii), the edges in such a cut could not all be deleted, so one of them lies in \( T \) . Proof of Theorem 8.5.10. The implication (ii) \( \rightarrow \) (i) follows from our three lemmas. For (i) \( \rightarrow \) (ii), let \( G \) have edge-disjoint topological spanning trees \( {T}_{1},\ldots ,{T}_{k} \), and consider a partition \( P \) of \( V\left( G\right) \) into \( \ell \) sets. If there are infinitely many cross-edges, there is nothing to show; so we assume there are only finitely many. For each \( i \in \{ 1,\ldots, k\} \), let \( {T}_{i}^{\prime } \) be the multigraph of order \( \ell \) which the edges of \( {T}_{i} \) induce on \( P \) . To establish that \( G \) has at least \( k\left( {\ell - 1}\right) \) cross-edges, we show that the graphs \( {T}_{i}^{\prime } \) are connected. If not, then some \( {T}_{i}^{\prime } \) has a vertex partition crossed by no edge of \( {T}_{i} \) . This partition induces a cut of \( G \) that contains no edge of \( {T}_{i} \) . By our assumption that \( G \) has only finitely many cross-edges, this cut is finite. By Lemma 8.5.5 (ii), this contradicts the arc-connectedness of \( {T}_{i} \) . ## Exercises 1. \( {}^{ - } \) Show that a connected graph is countable if all its vertices have countable degrees. 2. \( {}^{ - } \) Given countably many sequences \( {\sigma }^{i} = {s}_{1}^{i},{s}_{2}^{i},\ldots \left( {i \in \mathbb{N}}\right) \) of natural numbers, f

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

there is nothing to show; so we assume there are only finitely many. For each \( i \in \{ 1,\ldots, k\} \), let \( {T}_{i}^{\prime } \) be the multigraph of order \( \ell \) which the edges of \( {T}_{i} \) induce on \( P \) . To establish that \( G \) has at least \( k\left( {\ell - 1}\right) \) cross-edges, we show that the graphs \( {T}_{i}^{\prime } \) are connected. If not, then some \( {T}_{i}^{\prime } \) has a vertex partition crossed by no edge of \( {T}_{i} \) . This partition induces a cut of \( G \) that contains no edge of \( {T}_{i} \) . By our assumption that \( G \) has only finitely many cross-edges, this cut is finite. By Lemma 8.5.5 (ii), this contradicts the arc-connectedness of \( {T}_{i} \) . ## Exercises 1. \( {}^{ - } \) Show that a connected graph is countable if all its vertices have countable degrees. 2. \( {}^{ - } \) Given countably many sequences \( {\sigma }^{i} = {s}_{1}^{i},{s}_{2}^{i},\ldots \left( {i \in \mathbb{N}}\right) \) of natural numbers, find one sequence \( \sigma = {s}_{1},{s}_{2},\ldots \) that beats every \( {\sigma }^{i} \) eventually, i.e. such that for every \( i \) there exists an \( n\left( i\right) \) such that \( {s}_{n} > {s}_{n}^{i} \) for all \( n \geq n\left( i\right) \) . 3. Can a countable set have uncountably many subsets whose intersections have finitely bounded size? 4. \( {}^{ - } \) Let \( T \) be an infinite rooted tree. Show that every ray in \( T \) has an increasing tail, that is, a tail whose sequence of vertices increases in the tree-order associated with \( T \) and its root. 5. \( {}^{ - } \) Let \( G \) be an infinite graph and \( A, B \subseteq V\left( G\right) \) . Show that if no finite set of vertices separates \( A \) from \( B \) in \( G \), then \( G \) contains an infinite set of disjoint \( A - B \) paths. 6. \( {}^{ - } \) In Proposition 8.1.1, the existence of a spanning tree was proved using Zorn's lemma 'from below', to find a maximal acyclic subgraph. For finite graphs, one can also use induction 'from above', to find a minimal spanning connected subgraph. What happens if we apply Zorn's lemma 'from above' to find such a subgraph? 7. \( {}^{ - } \) Show that for every \( k \in \mathbb{N} \) there exists an infinitely connected graph of girth at least \( k \) . 8. Construct, for any given \( k \in \mathbb{N} \), a planar \( k \) -connected graph. Can you construct one whose girth is also at least \( k \) ? Can you construct an infinitely connected planar graph? 9. \( {}^{ - } \) Theorem 8.1.3 implies that there exists an \( \mathbb{N} \rightarrow \mathbb{N} \) function \( {f}_{\chi } \) such that, for every \( k \in \mathbb{N} \), every infinite graph of chromatic number at least \( {f}_{\chi }\left( k\right) \) has a finite subgraph of chromatic number at least \( k \) . (Namely, let \( {f}_{\chi } \) be the identity on \( \mathbb{N} \) .) Are there similar functions \( {f}_{\delta } \) and \( {f}_{\kappa } \) for the minimum degree and connectivity? 10. Prove Theorem 8.1.3 for countable graphs using the fact that, in this case, the topological space \( X \) defined in the second proof of the theorem is sequentially compact. (Thus, every infinite sequence of points in \( X \) has a convergent subsequence: there is an \( x \in X \) such that every neighbourhood of \( X \) contains a tail of the subsequence.) 11. \( {}^{ + } \) Show that, given \( k \in \mathbb{N} \) and an edge \( e \) in a graph \( G \), there are only finitely many bonds in \( G \) that consist of exactly \( k \) edges and contain \( e \) . 12. \( {}^{ - } \) Extend Theorem 2.4.4 to infinite graphs. 13. Rephrase Gallai's cycle-cocycle partition theorem (Ex. 35, Ch. 1) in terms of degrees, and extend the equivalent version to locally finite graphs. 14. Prove Theorem 8.4.8 for locally finite graphs. Does your proof extend to arbitrary countable graphs? 15. Extend the marriage theorem to locally finite graphs, but show that it fails for countable graphs with infinite degrees. 16. \( {}^{ + } \) Show that a locally finite graph \( G \) has a 1-factor if and only if, for every finite set \( S \subseteq V\left( G\right) \), the graph \( G - S \) has at most \( \left| S\right| \) odd (finite) components. Find a counterexample that is not locally finite. 17. \( {}^{ + } \) Extend Kuratowski’s theorem to countable graphs. 18. \( {}^{ - } \) A vertex \( v \in G \) is said to dominate an end \( \omega \) of \( G \) if any of the following three assertions holds; show that they are equivalent. (i) For some ray \( R \in \omega \) there is an infinite \( v - R \) fan in \( G \) . (ii) For every ray \( R \in \omega \) there is an infinite \( v - R \) fan in \( G \) . (iii) No finite subset of \( V\left( {G - v}\right) \) separates \( v \) from a ray in \( \omega \) . 19. Show that a graph \( G \) contains a \( T{K}^{{\aleph }_{0}} \) if and only if some end of \( G \) is dominated by infinitely many vertices. 20. Construct a countable graph with uncountably many thick ends. 21. Show that a countable tree has uncountably many ends if and only if it contains a subdivision of the binary tree \( {T}_{2} \) . 22. A graph \( G = \left( {V, E}\right) \) is called bounded if for every vertex labelling \( \ell : V \rightarrow \mathbb{N} \) there exists a function \( f : \mathbb{N} \rightarrow \mathbb{N} \) that exceeds the labelling along any ray in \( G \) eventually. (Formally: for every ray \( {v}_{1}{v}_{2}\ldots \) in \( G \) there exists an \( {n}_{0} \) such that \( f\left( n\right) > \ell \left( {v}_{n}\right) \) for every \( n > {n}_{0} \) .) Prove the following assertions: (i) The ray is bounded. (ii) Every locally finite connected graph is bounded. \( {\left( \mathrm{{iii}}\right) }^{ + } \) A countable tree is bounded if and only if it contains no subdivision of the \( {\aleph }_{0} \) -regular tree \( {T}_{{\aleph }_{0}} \) . 23. \( {}^{ + } \) Let \( T \) be a tree with root \( r \), and let \( \leq \) denote the tree-order on \( V\left( T\right) \) associated with \( T \) and \( r \) . Show that \( T \) contains no subdivision of the \( {\aleph }_{1} \) -regular tree \( {T}_{{\aleph }_{1}} \) if and only if \( T \) has an ordinal labelling \( t \mapsto o\left( t\right) \) such that \( o\left( t\right) \geq o\left( {t}^{\prime }\right) \) whenever \( t < {t}^{\prime } \) but no more than countably many vertices of \( T \) have the same label. 24. Show that a locally finite connected vertex-transitive graph has exactly \( 0,1,2 \) or infinitely many ends. 25. \( {}^{ + } \) Show that the automorphisms of a graph \( G = \left( {V, E}\right) \) act naturally on its ends, i.e., that every automorphism \( \sigma : V \rightarrow V \) can be extended to a map \( \sigma : \Omega \left( G\right) \rightarrow \Omega \left( G\right) \) such that \( \sigma \left( R\right) \in \sigma \left( \omega \right) \) whenever \( R \) is a ray in an end \( \omega \) . Prove that, if \( G \) is connected, every automorphism \( \sigma \) of \( G \) fixes a finite set of vertices or an end. If \( \sigma \) fixes no finite set of vertices, can it fix more than one end? More than two? 26. \( {}^{ - } \) Show that a locally finite spanning tree of a graph \( G \) contains a ray from every end of \( G \) . 27. A ray in a graph follows another ray if the two have infinitely many vertices in common. Show that if \( T \) is a normal spanning tree of \( G \) then every ray of \( G \) follows a unique normal ray of \( T \) . 28. Show that the following assertions are equivalent for connected countable graphs \( G \) . (i) \( G \) has a locally finite spanning tree. (ii) \( G \) has a locally finite normal spanning tree. (iii) Every normal spanning tree of \( G \) is locally finite. (iv) For no finite separator \( X \subseteq V\left( G\right) \) does \( G - X \) have infinitely many components. 29. Use the previous exercise to show that every (countable) planar 3- connected graph has a locally finite spanning tree. 30. Let \( G \) be a connected graph. Call a set \( U \subseteq V\left( G\right) \) dispersed if every ray in \( G \) can be separated from \( U \) by a finite set of vertices. (In the topology of Section 8.5, these are precisely the closed subsets of \( V\left( G\right) \) .) (i) Prove Jung’s theorem that \( G \) has a normal spanning tree if and only if \( V\left( G\right) \) is a countable union of dispersed sets. (ii) Deduce that if \( G \) has a normal spanning tree then so does every connected minor of \( G \) . 31. \( {}^{ - } \) Use Exercise 21 to prove that a countable graph with uncountably many ends has continuum many ends. 32. \( {}^{ + } \) Show that the vertices of any infinite connected locally finite graph can be enumerated in such a way that every vertex is adjacent to some later vertex. 33. (i) Prove that if a given end of a graph contains \( k \) disjoint rays for every \( k \in \mathbb{N} \) then it contains infinitely many disjoint rays. (ii) Prove that if a given end of a graph contains \( k \) edge-disjoint rays for every \( k \in \mathbb{N} \) then it contains infinitely many edge-disjoint rays. 34. \( {}^{ + } \) Prove that if a graph contains \( k \) disjoint double rays for every \( k \in \mathbb{N} \) then it contains infinitely many disjoint double rays. 35. Show that, in the ubiquity conjecture, the host graphs \( G \) considered can be assumed to be locally finite too. 36. Show that the modified comb below is not ubiquitous with respect to the subgraph relation. Does it become ubiquitous if we delete its 3-star on the left?  37. Show that if a graph contains infinitely many distinct cycles then it contains infinitely many edge-disjoint cycles. 38. Imitate the proof of Theorem 8.2.6 to find a function \( f : \mathbb{N} \rightarrow \mathbb{N} \) such that whenever an end \( \omega \) of a graph \( G \) cont

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

Prove that if a given end of a graph contains \( k \) edge-disjoint rays for every \( k \in \mathbb{N} \) then it contains infinitely many edge-disjoint rays. 34. \( {}^{ + } \) Prove that if a graph contains \( k \) disjoint double rays for every \( k \in \mathbb{N} \) then it contains infinitely many disjoint double rays. 35. Show that, in the ubiquity conjecture, the host graphs \( G \) considered can be assumed to be locally finite too. 36. Show that the modified comb below is not ubiquitous with respect to the subgraph relation. Does it become ubiquitous if we delete its 3-star on the left?  37. Show that if a graph contains infinitely many distinct cycles then it contains infinitely many edge-disjoint cycles. 38. Imitate the proof of Theorem 8.2.6 to find a function \( f : \mathbb{N} \rightarrow \mathbb{N} \) such that whenever an end \( \omega \) of a graph \( G \) contains \( f\left( k\right) \) disjoint rays there is a \( k \times \mathbb{N} \) grid in \( G \) whose rays all belong to \( \omega \) . 39. Show that there is no universal locally finite connected graph for the subgraph relation. 40. Construct a universal locally finite connected graph for the minor relation. Is there one for the topological minor relation? 41. \( {}^{ - } \) Show that each of the following operations performed on the Rado graph \( R \) leaves a graph isomorphic to \( R \) : (i) taking the complement, i.e. changing all edges into non-edges and vice versa; (ii) deleting finitely many vertices; (iii) changing finitely many edges into non-edges or vice versa; (iv) changing all the edges between a finite vertex set \( X \subseteq V\left( R\right) \) and its complement \( V\left( R\right) \smallsetminus X \) into non-edges, and vice versa. 42. \( {}^{ - } \) Prove that the Rado graph is homogeneous. 43. Show that a homogeneous countable graph is determined uniquely, up to isomorphism, by the class of (the isomorphism types of) its finite subgraphs. 44. Recall that subgraphs \( {H}_{1},{H}_{2},\ldots \) of a graph \( G \) are said to partition \( G \) if their edge sets form a partition of \( E\left( G\right) \) . Show that the Rado graph can be partitioned into any given countable set of countable locally finite graphs, as long as each of them contains at least one edge. 45. \( {}^{ - } \) A linear order is called dense if between any two elements there lies a third. (i) Find, or construct, a countable dense linear order that has neither a maximal nor a minimal element. (ii) Show that this order is unique, i.e. that every two such orders are order-isomorphic. (Definition?) (iii) Show that this ordering is universal among the countable linear orders. Is it homogeneous? (Supply appropriate definitions.) 46. Given a bijection \( f \) between \( \mathbb{N} \) and \( {\left\lbrack \mathbb{N}\right\rbrack }^{ < \omega } \), let \( {G}_{f} \) be the graph on \( \mathbb{N} \) in which \( u, v \in \mathbb{N} \) are adjacent if \( u \in f\left( v\right) \) or vice versa. Prove that all such graphs \( {G}_{f} \) are isomorphic. 47. (for set theorists) Show that, given any countable model of set theory, the graph whose vertices are the sets and in which two sets are adjacent if and only if one contains the other as an element, is the Rado graph. 48. Let \( G \) be a locally finite graph. Let us say that a finite set \( S \) of vertices separates two ends \( \omega \) and \( {\omega }^{\prime } \) if \( C\left( {S,\omega }\right) \neq C\left( {S,{\omega }^{\prime }}\right) \) . Use Proposition 8.4.1 to show that if \( \omega \) can be separated from \( {\omega }^{\prime } \) by \( k \in \mathbb{N} \) but no fewer vertices, then \( G \) contains \( k \) disjoint double rays each with one tail in \( \omega \) and one in \( {\omega }^{\prime } \) . Is the same true for all graphs that are not locally finite? 49. \( {}^{ + } \) Prove the following more structural version of Exercise 33 (i). Let \( \omega \) be an end of a countable graph \( G \) . Show that either \( G \) contains a \( T{K}^{{\aleph }_{0}} \) with all its rays in \( \omega \), or there are disjoint finite sets \( {S}_{0},{S}_{1},{S}_{2},\ldots \) such that \( \left| {S}_{1}\right| \leq \left| {S}_{2}\right| \leq \ldots \) and, with \( {C}_{i} \mathrel{\text{:=}} C\left( {{S}_{0} \cup {S}_{i},\omega }\right) \), we have for all \( i < j \) that \( {C}_{i} \supseteq {C}_{j} \) and \( {G}_{i} \mathrel{\text{:=}} G\left\lbrack {{S}_{i} \cup {C}_{i}}\right\rbrack \) contains \( \left| {S}_{i}\right| \) disjoint \( {S}_{i} - {S}_{i + 1} \) paths. 50. Construct an example of a small limit of large waves. 51. \( {}^{ + } \) Prove Theorem 8.4.2 for trees. 52. \( {}^{ + } \) Prove Pym’s theorem (8.4.7). 53. (i) \( {}^{ - } \) Prove the naive extension of Dilworth’s theorem to arbitrary infinite posets \( P \) : if \( P \) has no antichain of order \( k \in \mathbb{N} \), then \( P \) can be partitioned into fewer than \( k \) chains. (A proof for countable \( P \) will do.) (ii) \( {}^{ - } \) Find a poset that has no infinite antichain and no partition into finitely many chains. (iii) For posets without infinite chains, deduce from Theorem 8.4.8 the following Erdős-Menger-type extension of Dilworth's theorem: every such poset has a partition \( \mathcal{C} \) into chains such that some antichain meets all the chains in \( \mathcal{C} \) . 54. Let \( G \) be a countable graph in which for every partial matching there is an augmenting path. Let \( M \) be any matching. Is there a sequence, possibly transfinite, of augmenting paths (each for the then current matching) that turns \( M \) into a 1 -factor? 55. Find an uncountable graph in which every partial matching admits an augmenting path but which has no 1-factor. 56. Construct a locally finite factor-critical graph (or prove that none exists). 57. \( {}^{ - } \) Let \( G \) be a countable graph whose finite subgraphs are all perfect. Show that \( G \) is weakly perfect but not necessarily perfect. 58. \( {}^{ + } \) Let \( G \) be the incomparability graph of the binary tree. (Thus, \( V\left( G\right) = \) \( V\left( {T}_{2}\right) \), and two vertices are adjacent if and only if they are incomparable in the tree-order of \( {T}_{2} \) .) Show that \( G \) is perfect but not strongly perfect. 59. Let \( G \) be a graph, \( X \subseteq V\left( G\right) \), and \( R \in \omega \in \Omega \left( G\right) \) . Show that \( G \) contains a comb with spine \( R \) and teeth in \( X \) if and only if \( \omega \in \bar{X} \) . 60. Give an independent proof of Proposition 8.5.1 using sequential compactness and the infinity lemma. 61. \( {}^{ + } \) Let \( G \) be a connected countable graph that is not locally finite. Show that \( \left| G\right| \) is not compact, but that \( \Omega \left( G\right) \) is compact if and only if for every finite set \( S \subseteq V\left( G\right) \) only finitely many components of \( G - S \) contain a ray. 62. Given graphs \( H \subseteq G \), let \( \eta : \Omega \left( H\right) \rightarrow \Omega \left( G\right) \) assign to every end of \( H \) the unique end of \( G \) containing it as a subset (of rays). For the following questions, assume that \( H \) is connected and \( V\left( H\right) = V\left( G\right) \) . (i) Show that \( \eta \) need not be injective. Must it be surjective? (ii) Investigate how \( \eta \) relates the subspace \( \Omega \left( H\right) \) of \( \left| H\right| \) to its image in \( \left| G\right| \) . Is \( \eta \) always continuous? Is it open? Do the answers to these questions change if \( \eta \) is known to be injective? (iii) A spanning tree is called end-faithful if \( \eta \) is bijective, and topologically end-faithful if \( \eta \) is a homeomorphism. Show that every connected countable graph has a topologically end-faithful spanning tree. 63. \( {}^{ + } \) Let \( G \) be a connected graph. Assuming that \( G \) has a normal spanning tree, define a metric on \( \left| G\right| \) that induces its usual topology. Conversely, use Jung’s theorem of Exercise 30 to show that if \( V \cup \Omega \subseteq \left| G\right| \) is metrizable then \( G \) has a normal spanning tree. 64. \( {}^{ + } \) (for topologists) In a locally compact, connected, and locally connected Hausdorff space \( X \), consider sequences \( {U}_{1} \supseteq {U}_{2} \supseteq \ldots \) of open, nonempty, connected subsets with compact frontiers such that \( \mathop{\bigcap }\limits_{{i \in \mathbb{N}}}\overline{{U}_{i}} = \varnothing \) . Call such a sequence equivalent to another such sequence if every set of one sequence contains some set of the other, and vice versa. Note that this is indeed an equivalence relation, and call its classes the Freudenthal ends of \( X \) . Now add these to the space \( X \), and define a natural topology on the extended space \( \widehat{X} \) that makes it homeomorphic to \( \left| X\right| \) if \( X \) is a graph, by a homeomorphism that is the identity on \( X \) . 65. Let \( F \) be a set of edges in a locally finite graph \( G \), and let \( A \mathrel{\text{:=}} \overline{\bigcup F} \) be its closure in \( \left| G\right| \) . Show that \( F \) is a circuit if and only if, for every two edges \( e,{e}^{\prime } \in F \), the set \( A \smallsetminus \mathring{e} \) is connected but \( A \smallsetminus \left( {\mathring{e} \cup {\mathring{e}}^{\prime }}\right) \) is disconnected in \( \left| G\right| \) . 66. Does every infinite locally finite 2-connected graph contain an infinite circuit? Does it contain an infinite bond? 67. Show that the union of all the edges contained in an arc or circle \( C \) in \( \left| G\right| \) is dense in \( C \) . 68. Let \( T \) be a spanning tree of a graph \( G \) . Note that \( \bar{T} \) is a connected subset of \( \left| G\right| \) . Without using Lemma 8.5.4, show that if

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

\) that makes it homeomorphic to \( \left| X\right| \) if \( X \) is a graph, by a homeomorphism that is the identity on \( X \) . 65. Let \( F \) be a set of edges in a locally finite graph \( G \), and let \( A \mathrel{\text{:=}} \overline{\bigcup F} \) be its closure in \( \left| G\right| \) . Show that \( F \) is a circuit if and only if, for every two edges \( e,{e}^{\prime } \in F \), the set \( A \smallsetminus \mathring{e} \) is connected but \( A \smallsetminus \left( {\mathring{e} \cup {\mathring{e}}^{\prime }}\right) \) is disconnected in \( \left| G\right| \) . 66. Does every infinite locally finite 2-connected graph contain an infinite circuit? Does it contain an infinite bond? 67. Show that the union of all the edges contained in an arc or circle \( C \) in \( \left| G\right| \) is dense in \( C \) . 68. Let \( T \) be a spanning tree of a graph \( G \) . Note that \( \bar{T} \) is a connected subset of \( \left| G\right| \) . Without using Lemma 8.5.4, show that if \( T \) is locally finite then \( \bar{T} \) is arc-connected. Find an example where \( \bar{T} \) is not arc-connected. 69. Prove that the circle shown in Figure 8.5.1 is really a circle, by exhibiting a homeomorphism with \( {S}^{1} \) . 70. Deduce Lemma 8.5.6 from Lemma 8.5.4. 71. Let \( G \) be a connected locally finite graph. Show that the following assertions are equivalent for a spanning subgraph \( T \) of \( G \) : (i) \( \bar{T} \) is a topological spanning tree of \( \left| G\right| \) ; (ii) \( T \) is edge-maximal such that \( \bar{T} \) contains no circle; (iii) \( T \) is edge-minimal with \( \bar{T} \) arc-connected. 72. \( {}^{ - } \) Observe that a topological spanning tree need not be homeomorphic to a tree. Is it homeomorphic to the space \( \left| T\right| \) for a suitable tree \( T \) ? 73. Show that connected graphs with only one end have topological spanning trees. 74. \( {}^{ + } \) Let \( G \) be a locally finite graph and \( X \) a standard subspace of \( \left| G\right| \) . Prove that arc-components \( A \) of \( X \) are closed in \( X \) . Deduce that the fundamental cuts of any topological spanning tree of \( G \) are finite. 75. To show that Theorem 3.2.3 does not generalize to infinite graphs with the 'finite' cycle space as defined in Chapter 1.9, construct a 3- connected locally finite planar graph with a separating cycle that is not a finite sum of non-separating induced cycles. Can you find an example where even infinite sums of finite non-separating induced cycles do not generate all separating cycles? 76. \( {}^{ - } \) As a converse to Theorem 8.5.8 (iii), show that the fundamental circuits of an ordinary spanning tree \( T \) of a locally finite graph \( G \) do not generate \( \mathcal{C}\left( G\right) \) unless \( \bar{T} \) is a topological spanning tree. 77. Prove that the edge set of a countable graph \( G \) can be partitioned into finite circuits if \( G \) has no odd cut. Where does your argument break down if \( G \) is uncountable? 78. Explain why Theorem 8.5.8 (ii) is needed in the proof of Corollary 8.5.9: can’t we just combine the constituent sums of circuits for the \( {D}_{i} \) (from our assumption that \( \left. {{D}_{i} \in \mathcal{C}\left( G\right) }\right) \) into one big family? If not, can you still prove the same statement without appealing to Theorem 8.5.8 (ii)? 79. \( {}^{ + } \) Call a continuous (but not necessarily injective) map \( \sigma : {S}^{1} \rightarrow \left| G\right| \) a topological Euler tour of \( G \) if every inner point of an edge of \( G \) is the image of exactly one point of \( {S}^{1} \) . (Thus, every edge is traversed exactly once, and in a 'straight' manner.) Use Theorem 8.5.8 (ii) to show that \( G \) admits a topological Euler tour if and only if \( G \) is connected and \( E\left( G\right) \in \mathcal{C}\left( G\right) \) 80. \( {}^{ + } \) An open Euler tour in an infinite graph \( G \) is a 2-way infinite walk \( \ldots {e}_{-1}{v}_{0}{e}_{0}\ldots \) that contains every edge of \( G \) exactly once. Show that \( G \) contains an open Euler tour if and only if \( G \) is countable, \( G \) is connected, every vertex has even or infinite degree, and any finite cut \( F = E\left( {{V}_{1},{V}_{2}}\right) \) with both \( {V}_{1} \) and \( {V}_{2} \) infinite is odd. ## Notes There is no comprehensive monograph on infinite graph theory, but over time several surveys have been published. A relatively wide-ranging collection of survey articles can be found in R. Diestel (ed.), Directions in Infinite Graph Theory and Combinatorics, North-Holland 1992. (This has been reprinted as Volume 95 of the journal Discrete Mathematics.) Some of the articles there address purely graph-theoretic aspects of infinite graphs, while others point to connections with other fields in mathematics such as differential geometry, topological groups, or logic. A survey of infinite graph theory as a whole was given by C. Thomas-sen, Infinite graphs, in (L.W. Beineke & R.J. Wilson, eds.) Selected Topics in Graph Theory 2, Academic Press 1983. This also treats a number of aspects of infinite graph theory not considered in our chapter here, including problems of Erdős concerning infinite chromatic number, infinite Ramsey theory (also known as partition calculus), and reconstruction. The first two of these topics receive much attention also in A. Hajnal's chapter of the Handbook of Combinatorics (R.L. Graham, M. Grötschel & L. Lovász, eds.), North-Holland 1995, which has a strong set-theoretical flavour. (See the end of these notes for more references in this direction.) A specific survey on reconstruction by Nash-Williams can be found in the Directions volume cited above. A relatively recent collection of various unsolved problems is offered in R. Halin, Miscellaneous problems on infinite graphs, J. Graph Theory 35 (2000), 128-151. A good general reference for infinite graphs (as well as finite) is R. Halin, Graphentheorie (2nd ed.), Wissenschaftliche Buchgesellschaft 1989. A more specific monograph on the theory of simplicial decompositions (see Chapter 12) is R. Diestel, Graph Decompositions, Oxford University Press 1990. Chapter 12.4 closes with a few theorems about forbidden minors in infinite graphs. Infinite graph theory has a number of interesting individual results which, as yet, stand essentially by themselves. One such is a theorem of A. Huck, F. Niedermeyer and S. Shelah, Large \( \kappa \) -preserving sets in infinite graphs, J. Graph Theory 18 (1994), 413-426, which says that every infinitely connected graph \( G \) has a set \( S \) of \( \left| G\right| \) vertices such that \( \kappa \left( {G - {S}^{\prime }}\right) = \kappa \left( G\right) \) for every \( {S}^{\prime } \subseteq S \) . Another is Halin’s bounded graph conjecture, which characterizes the bounded graphs by four forbidden substructures. (See Exercise 22 (iii) for the definition of 'bounded' and the tree case of the conjecture.) A proof can be found in R. Diestel & I.B. Leader, A proof of the bounded graph conjecture, Invent. math. 108 (1992), 131-162. König's infinity lemma, or König's lemma for short, is as old as the first-ever book on graph theory, which includes it: D. König, Theorie der endlichen und unendlichen Graphen, Akademische Verlagsgesellschaft, Leipzig 1936. In addition to this and Tychonoff's theorem, compactness proofs can also come in the following two guises (see Hajnal's Handbook chapter): as applications of Rado's selection lemma, or of Gödel's compactness theorem from first-order logic. Both are logically equivalent to Tychonoff's theorem; the choice of which to use is more a matter of familiarity with one terminology or the other than of any material importance. Theorem 8.1.3 is due to N. G. de Bruijn and P. Erdős, A colour problem for infinite graphs and a problem in the theory of relations, Indag. Math. 13 (1951), 371-373. Unlike for the chromatic number, a bound on the colouring number of all finite subgraphs does not extend to the whole graph by compactness. P. Erdős & A. Hajnal, On the chromatic number of graphs and set systems, Acta Math. Acad. Sci. Hung. 17 (1966), 61-99, proved that if every finite subgraph of \( G \) has colouring number at most \( k \) then \( G \) has colouring number at most \( {2k} - 2 \), and showed that this is best possible. The unfriendly partition conjecture is one of the best-known open problems in infinite graph theory, but there are few results. E.C.Milner and S. Shelah, Graphs with no unfriendly partitions, in (A. Baker, B. Bollobás & A. Hajnal, eds.), A tribute to Paul Erdős, Cambridge University Press 1990, construct an uncountable counterexample, but show that every graph has an unfriendly partition into three classes. (The original conjecture, which they attribute to R. Cowan and W. Emerson (unpublished), appears to have asserted for every graph the existence of a vertex partition into any given finite number of classes such that every vertex has at least as many neighbours in other classes as in its own.) Some positive results for bipartitions were obtained by R. Aharoni, E.C. Milner and K. Prikry, Unfriendly partitions of graphs, J. Combin. Theory B 50 (1990), 1-10. Theorem 8.2.4 is a special case of the result stated in Exercise 30 (i), which is due to H.A. Jung, Wurzelbäume und unendliche Wege in Graphen, Math. Nachr. 41 (1969), 1-22. The graphs that admit a normal spanning tree can be characterized by forbidden minors: as shown in R. Diestel & I. Leader, Normal spanning trees, Aronszajn trees and excluded minors, J. London Math. Soc. 63 (2001), 16-32, there are two types of graphs that are easily seen not to have normal spanning trees, and one of these must occur as a minor in every graph without a normal spanning tree. Note that such a characterization is possible only because the class of graphs admitting a normal spanning tree is closed under taking connected minors - a consequence of

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

lasses such that every vertex has at least as many neighbours in other classes as in its own.) Some positive results for bipartitions were obtained by R. Aharoni, E.C. Milner and K. Prikry, Unfriendly partitions of graphs, J. Combin. Theory B 50 (1990), 1-10. Theorem 8.2.4 is a special case of the result stated in Exercise 30 (i), which is due to H.A. Jung, Wurzelbäume und unendliche Wege in Graphen, Math. Nachr. 41 (1969), 1-22. The graphs that admit a normal spanning tree can be characterized by forbidden minors: as shown in R. Diestel & I. Leader, Normal spanning trees, Aronszajn trees and excluded minors, J. London Math. Soc. 63 (2001), 16-32, there are two types of graphs that are easily seen not to have normal spanning trees, and one of these must occur as a minor in every graph without a normal spanning tree. Note that such a characterization is possible only because the class of graphs admitting a normal spanning tree is closed under taking connected minors - a consequence of Jung's theorem (see Exercise 30(ii)) for which, oddly, no direct proof is known. One corollary of the characterization is that a connected graph has a normal spanning tree if and only if all its minors have countable colouring number. Theorems 8.2.5 and 8.2.6 are from R. Halin, Über die Maximalzahl frem-der unendlicher Wege, Math. Nachr. 30 (1965), 63-85. Our proof of Theorem 8.2.5 is due to Andreae (unpublished); our proof of Theorem 8.2.6 is new. Halin's paper also includes a structure theorem for graphs that do not contain infinitely many disjoint rays. Except for a finite set of vertices, such a graph can be written as an infinite chain of rayless subgraphs each overlapping the previous in exactly \( m \) vertices, where \( m \) is the maximum number of disjoint rays (which exists by Theorem 8.2.5). These overlap sets are disjoint, and there are \( m \) disjoint rays containing exactly one vertex from each of them. A good reference on ubiquity, including the ubiquity conjecture, is Th. Andreae, On disjoint configurations in infinite graphs, J. Graph Theory 39 \( \left( {2002}\right) ,{222} - {229} \) . Universal graphs have been studied mostly with respect to the induced subgraph relation, with numerous but mostly negative results. See G. Cherlin, S. Shelah & N. Shi, Universal graphs with forbidden subgraphs and algebraic closure, Adv. Appl. Math. 22 (1999), 454-491, for an overview and a model-theoretic framework for the proof techniques typically applied. The Rado graph is probably the best-studied single graph in the graph theory literature (with the Petersen graph a close runner-up). The most comprehensive source for anything related to it (and far beyond) is R. Fraïssé, Theory of Relations (2nd edn.), Elsevier 2000. More accessible introductions are given by N. Sauer in his appendix to Fraïssé's book, and by P.J. Cameron, The random graph, in (R.L. Graham & J. Nešetřil, eds.): The Mathematics of Paul Erdős, Springer 1997, and its references. Theorem 8.3.1 is due to P. Erdős and A. Rényi, Asymmetric graphs, Acta Math. Acad. Sci. Hung. 14 (1963), 295-315. The existence part of their proof is probabilistic and will be given in Theorem 11.3.5. Rado's explicit definition of the graph \( R \) was given in R. Rado, Universal graphs and universal functions, Acta Arithm. 9 (1964),393-407. However, its universality and that of \( {R}^{r} \) are already included in more general results of B. Jónsson, Universal relational systems, Math. Scand. 4 (1956), 193-208. Theorem 8.3.3 is due to A.H. Lachlan and R.E. Woodrow, Countable ul-trahomogeneous undirected graphs, Trans. Amer. Math. Soc. 262 (1980), 51- 94. The classification of the countable homogeneous directed graphs is much more difficult still. It was achieved by G. Cherlin, The classification of countable homogeneous directed graphs and countable homogeneous \( n \) -tournaments, Mem. Am. Math. Soc. 621 (1998), which also includes a shorter proof of Theorem 8.3.3. Proposition 8.3.2, too, has a less trivial directed analogue: the countable directed graphs that are isomorphic to at least one of the two sides induced by any bipartition of their vertex set are precisely the edgeless graph, the random tournament, the transitive tournaments of order type \( {\omega }^{\alpha } \), and two specific orientations of the Rado graph (R. Diestel, I. Leader, A. Scott & S. Thomassé, Partitions and orientations of the Rado graph, Trans. Amer. Math. Soc. (to appear). Theorem 8.3.4 is proved in R.Diestel & D.Kühn, A universal planar graph under the minor relation, J. Graph Theory 32 (1999), 191-206. It is not known whether or not there is a universal planar graph for the topological minor relation. However it can be shown that there is no minor-universal graph for embeddability in any closed surface other than the sphere; see the above paper. When Erdős conjectured his extension of Menger's theorem is not known; C.St.J.A. Nash-Williams, Infinite graphs - a survey, J. Combin. Theory B 3 (1967), 286-301, cites the proceedings of a 1963 conference as its source. Its proof as Theorem 8.4.2 by Aharoni and Berger, Menger's theorem for infinite graphs (preprint 2005), came as the culmination of a long effort over many years, for the most part also due to Aharoni. Our proof of its countable case is adapted from R. Aharoni, Menger’s theorem for countable graphs, J. Combin. Theory B 43 (1987), 303-313. Theorem 8.4.2 can be extended to ends, as follows. Given two sets \( A, B \subseteq \) \( V\left( G\right) \cup \Omega \left( G\right) \), let us say that \( G \) satisfies the Erdős-Menger conjecture for \( A \) and \( B \) if \( G \) contains a set \( \mathcal{P} \) of paths (finite or infinite) whose closures in the space \( \left| G\right| \) defined in Section 8.5 are disjoint arcs each linking a point of \( A \) to a point of \( B \), and there is a set \( X \) consisting of one vertex or end from each path in \( \mathcal{P} \) such that every path in \( G \) whose closure links a point of \( A \) to one of \( B \) has a vertex or end in \( X \) . (Note that if \( A, B \subseteq V\left( G\right) \) then this statement coincides with Theorem 8.4.2.) Then every graph \( G \) satisfies the Erdős-Menger conjecture for all sets \( A, B \subseteq V\left( G\right) \cup \Omega \left( G\right) \) satisfying \( A \cap \bar{B} = \varnothing = \bar{A} \cap B \) , and there are counterexamples when this condition is violated. See H. Bruhn, R. Diestel & M. Stein, Menger's theorem for infinite graphs with ends, J. Graph Theory (to appear). There is also a purely topological version of the Erdős-Menger conjecture that asks for any set of disjoint \( A - B \) arcs in \( \left| G\right| \) together with a selection \( X \) of points, one from each of these arcs, that meets every \( A - B \) arc in \( \left| G\right| \) . An example of Kühn shows that this version of the Erdős-Menger conjecture can fail if \( \bar{A} \cap \bar{B} \neq \varnothing \) . However if we assume that \( \bar{A} \cap \bar{B} = \varnothing \), then the separator \( X \) provided by the theorem stated at the end of the last paragraph can be shown to meet every \( A - B \) arc in \( \left| G\right| \), not only those that are paths or closures of rays or double rays. Thus, the theorem cited above implies the purely topological version of the Erdős-Menger conjecture too. Theorem 8.4.7 is due to J.S. Pym, A proof of the linkage theorem, J. Math. Anal. Appl. 27 (1969), 636-638. The short proof outlined in Exercise 52 can be found in R. Diestel & C. Thomassen, A Cantor-Bernstein theorem for paths in graphs, Amer. Math. Monthly (to appear). The matching theorems of Chapter 2-König's duality theorem, Hall's marriage theorem, Tutte's 1-factor theorem, and the Gallai-Edmonds matching theorem extend essentially unchanged to locally finite graphs by compactness; see e.g. Exercises 14-16. For non-locally-finite graphs, matching theory is considerably deeper. A good survey and open problems can be found in R. Aharoni, Infinite matching theory, in the Directions volume cited earlier. A thorough account is given in M. Holz, K.P. Podewski & K. Steffens, Injective choice functions, Lecture Notes in Mathematics 1238 Springer-Verlag 1987. Most of the results and techniques for infinite matching were developed first for countable graphs, by Podewski and Steffens in the 1970s. In the 1980s, Aharoni extended them to arbitrary graphs, where things are more difficult still and additional methods are required. Theorem 8.4.8 is due to R. Aharoni, König's duality theorem for infinite bipartite graphs, J. London Math. Soc. 29 (1984), 1-12. The proof builds on R. Aharoni, C.St.J.A. Nash-Willaims & S. Shelah, A general criterion for the existence of transversals, Proc. London Math. Soc. 47 (1983), 43-68, and is described in detail in the book of Holz, Podewski and Steffens. Theorem 8.4.10 can be derived from the material in K. Steffens, Matchings in countable graphs, Can. J. Math. 29 (1977), 165-168. Theorem 8.4.11 is due to R. Aharoni, Matchings in infinite graphs, J. Com-bin. Theory B 44 (1988), 87-125; a shorter proof was given by Niedermeyer and Podewski, Matchable infinite graphs, J. Combin. Theory B 62 (1994), 213-227. The theorem was extended to \( f \) -factors by F. Niedermeyer, \( f \) -optimal factors of infinite graphs, also in the Directions volume cited earlier. The topology on \( G \) introduced in Section 8.5 coincides, when \( G \) is locally finite, with the usual topology of a 1-dimensional CW-complex. Then \( \left| G\right| \) can be interpreted as the compactification of \( G \) suggested by H. Freudenthal, Über die Enden topologischer Räume und Gruppen, Math. Zeit. 33 (1931), 692-713; see Exercise 64. For graphs that are not locally finite, the graph-theoretical notion of an end is more general than the topological one; see R. Diestel & D. Kühn, Graph-theoretical versus topological ends of graphs, J. Combin. Theory B 87 (2003), 197-206.

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29 (1977), 165-168. Theorem 8.4.11 is due to R. Aharoni, Matchings in infinite graphs, J. Com-bin. Theory B 44 (1988), 87-125; a shorter proof was given by Niedermeyer and Podewski, Matchable infinite graphs, J. Combin. Theory B 62 (1994), 213-227. The theorem was extended to \( f \) -factors by F. Niedermeyer, \( f \) -optimal factors of infinite graphs, also in the Directions volume cited earlier. The topology on \( G \) introduced in Section 8.5 coincides, when \( G \) is locally finite, with the usual topology of a 1-dimensional CW-complex. Then \( \left| G\right| \) can be interpreted as the compactification of \( G \) suggested by H. Freudenthal, Über die Enden topologischer Räume und Gruppen, Math. Zeit. 33 (1931), 692-713; see Exercise 64. For graphs that are not locally finite, the graph-theoretical notion of an end is more general than the topological one; see R. Diestel & D. Kühn, Graph-theoretical versus topological ends of graphs, J. Combin. Theory B 87 (2003), 197-206. Topological aspects of the subspaces \( \Omega \) and \( V \cup \Omega \) were studied extensively by Polat; see e.g. N. Polat, Ends and multi-endings I & II, J. Combin. Theory B 67 (1996), 56-110. The usual notion of an \( x - y \) path in a topological space \( X \) is that of a continuous (but not necessarily injective) map from \( \left\lbrack {0,1}\right\rbrack \) to \( X \) that maps 0 to \( x \) and 1 to \( y \) . One can show that the image of an \( x - y \) path in a Hausdorff space always contains an \( x - y \) arc - in particular, arc-connectedness is the same as the more common topological notion of path-connectedness - so it is largely a matter of convenience which of the two notions to consider. In the context of graphs it seems best to consider arcs: not only because topological paths could be confused with graph-theoretical paths, but also because the latter are 'injective' by definition, and are hence best generalized by arcs. A locally finite graph \( G \) for which \( \left| G\right| \) has a connected subset that is not arc-connected has been constructed by A. Georgakopoulos, Connected but not path-connected subspaces of infinite graphs, preprint 2005. A proof that closed connected subsets of \( \left| G\right| \) are arc-connected (Lemma 8.5.4) is given in R. Diestel & D. Kühn, Topological paths, cycles and spanning trees in infinite graphs, Europ. J. Combinatorics 25 (2004), 835-862. The (combinatorial) vertex-degree of an end is traditionally known as its multiplicity. The term 'degree', as well as its topological counterpart based on arcs, was introduced by H. Bruhn and M. Stein, On end degrees and infinite circuits in locally finite graphs (preprint 2004). Their paper includes proofs that the maxima in the definitions of topological end degrees are attained, that the topological degrees of the ends of \( G \) taken in the entire space \( \left| G\right| \) coincide with their combinatorial degrees, and of Lemma 8.5.6. Their main result is that the entire edge set of a locally finite graph lies in its cycle space if and only if every vertex and every end has even degree, with an appropriate division of the ends of infinite degree into 'even' and 'odd'. They conjecture that, like Proposition 1.9.2, this equivalence should extend to arbitrary sets \( F \subseteq E\left( G\right) \), with topological edge-degrees of ends. An interesting new aspect of end degrees is that they could make it possible to study extremal-type problems for infinite graphs that would otherwise make sense only for finite graphs. For example, while finite graphs of large enough minimum degree contain any desired topological minor or minor (see Chapter 7), an infinite graph of large minimum degree can be a tree. The ends of a tree, however, have degree 1 . An assumption that the degrees of both vertices and ends of an infinite graph are large can still not force a non-planar minor (because such graphs can be planar), but it might force arbitrarily highly connected subgraphs. Another approach to 'extremal' infinite graph theory, which seeks to force infinite substructures by assuming a lower bound for \( \begin{Vmatrix}{G\left\lbrack {{v}_{1},\ldots {v}_{n}}\right\rbrack }\end{Vmatrix} \) when \( V\left( G\right) = \left\{ {{v}_{1},{v}_{2},\ldots }\right\} \), is taken by J. Czipszer, P. Erdős and A. Hajnal, Some extremal problems on infinite graphs, Publ. Math. Inst. Hung. Acad. Sci., Ser. A 7 (1962), 441-457. For graphs \( G \) that are not locally finite, it can be natural to consider a coarser topology on \( \left| G\right| \), obtained by taking as basic open sets \( {\widehat{C}}_{\epsilon }\left( {S,\omega }\right) \) only those with \( \epsilon = 1 \) . Under this topology, \( \left| G\right| \) is no longer Hausdorff, because every vertex dominating an end \( \omega \) will lie in the closure of every \( \widehat{C}\left( {S,\omega }\right) \) . But \( \left| G\right| \) can now be compact, and it can have a natural quotient space-in which ends are identified with vertices dominating them and rays converge to vertices - that is both Hausdorff and compact. For details see R. Diestel, On end spaces and spanning trees (preprint 2004), where also Theorem 8.5.2 is proved. A proof of Lemma 8.5.3 can be found in \( §{47} \) of K. Kuratowski, Topology II, Academic Press 1968. Unlike the cycle space, the cut space \( {\mathcal{C}}^{ * }\left( G\right) \) of an infinite graph \( G \) can be defined as for finite graphs. It then contains infinite as well as finite cuts (which makes it a suitable partner of the cycle space, e.g. for plane duality), but this does not affect the proofs of its basic properties: it is still generated by the cuts of the form \( E\left( v\right) \) (Proposition 1.9.3); it consists of precisely those sets of edges that meet every finite circuit in an even number of edges (Ex. 30, Ch. 1); and every cut is a disjoint union of bonds (Proposition 1.9.4). Our topological notion of the cycle space \( \mathcal{C}\left( G\right) \) may appear natural in an infinite setting, but historically it is very young. It was developed in order to extend the classical applications of the cycle space of finite graphs, such as in planarity and duality, to locally finite graphs. As in the case of the tree-packing theorem (Theorem 8.5.10), those extensions fail when only finite circuits and sums are permitted, but they do hold for topological cycle spaces. Examples include Tutte's theorem (3.2.3) that the non-separating induced cycles generate the whole cycle space; MacLane's (4.5.1), Kelmans's (4.5.2) and Whitney's (4.6.3) characterizations of planarity; and Gallai's cycle-cocycle partition theorem (Ex. 35, Ch. 1). An expository account of examples and ideas that led to the topological definition of \( \mathcal{C}\left( G\right) \) is given in R. Diestel, The cycle space of an infinite graph, Combinatorics, Probability and Computing 14 (2005),59-79. These show that \( \mathcal{C}\left( G\right) \) is not unnecessarily complicated, in that no smaller collection of circuits suffices to generalize even the most basic facts about the cycle space of a finite graph. It also gives a survey of applications of \( \mathcal{C}\left( G\right) \) and of open problems, as well as references for all the results of Section 8.5 other than Theorem 8.5.10 (which is new). For graphs that are not locally finite, the problem of how best to define their cycle space is still far from solved. Theorem 8.5.8 is from R. Diestel & D. Kühn, On infinite cycles I-II, Com-binatorica 24 (2004), 69-116. Our proof of part (ii) via Lemma 8.5.4 was inspired by A. Vella, A fundamentally topological perspective on graph theory, PhD thesis, Waterloo 2004. Its corollary that locally finite graphs without odd cuts have edge-partitions into finite circuits easily extends to arbitrary countable graphs (Exercise 77), and is true even for uncountable graphs. This is a difficult theorem of C.St.J.A. Nash-Williams, Decomposition of graphs into closed and endless chains, Proc. London Math. Soc. 10 (1960), 221-238. Lacking the concept of an infinite circuit as we defined it here, Nash-Williams also sought to generalize the above and other theorems about finite cycles by replacing 'cycle' with '2-regular connected graph' (which may be finite or infinite). The resulting statements are not always as smooth as the finite theorems they generalize, but some substantial work has been done in this direction. C.St.J.A. Nash-Williams, Decompositions of graphs into two-way infinite paths, Can. J. Math. 15 (1963), 479-485, characterizes the graphs admitting edge-decompositions into double rays. F. Laviolette, Decompositions of infinite graphs I-II, J. Combin. Theory B 94 (2005), 259-333, characterizes the graphs admitting edge-decompositions into cycles and double rays. Results on the existence of spanning rays or double rays are referenced in the notes for Chapter 10. Topological spanning trees were introduced by R. Diestel and D. Kühn, Topological paths, cycles and spanning trees in infinite graphs, Europ. J. Combinatorics 25 (2004), 835-862. They are essential for the infinite tree packing theorem: if we replace them by ordinary spanning trees, Theorem 8.5.10 becomes false. This was shown by J.G. Oxley, On a packing problem for infinite graphs and independence spaces, J. Combin. Theory B 26 (1979), 123-130, disproving Nash-Williams's conjecture that the finite theorem should extend verbatim. What Tutte thought about an infinite version of the tree packing theorem is not recorded: in his original paper he treats the infinite case by defining 'semiconnected' subgraphs and proving Lemma 8.5.11, and leaves things at that. The companion to the finite tree-packing theorem, Nash-Williams's Theorem 2.4.4 that the edges of a graph can be covered by \( k \) forests if no set of \( \ell \) vertices spans more than \( k\left( {\ell - 1}\right) \) edges, extends easily by compactness (Exercise 1

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duced by R. Diestel and D. Kühn, Topological paths, cycles and spanning trees in infinite graphs, Europ. J. Combinatorics 25 (2004), 835-862. They are essential for the infinite tree packing theorem: if we replace them by ordinary spanning trees, Theorem 8.5.10 becomes false. This was shown by J.G. Oxley, On a packing problem for infinite graphs and independence spaces, J. Combin. Theory B 26 (1979), 123-130, disproving Nash-Williams's conjecture that the finite theorem should extend verbatim. What Tutte thought about an infinite version of the tree packing theorem is not recorded: in his original paper he treats the infinite case by defining 'semiconnected' subgraphs and proving Lemma 8.5.11, and leaves things at that. The companion to the finite tree-packing theorem, Nash-Williams's Theorem 2.4.4 that the edges of a graph can be covered by \( k \) forests if no set of \( \ell \) vertices spans more than \( k\left( {\ell - 1}\right) \) edges, extends easily by compactness (Exercise 12). However, in the infinite case it seems natural to ask for more: that the forests also have 'acirclic' closures. Suprisingly, perhaps, the assumption that no set of \( \ell \) vertices spans more than \( k\left( {\ell - 1}\right) \) edges does not imply that the edges of \( G \) (locally finite) can be covered by \( k \) such topological forests. However, if we assume in addition that every end of \( G \) has degree less than \( {2k} \) , then such a cover was shown to exist by M. Stein, Arboricity and tree-packing in locally finite graphs, preprint 2004. Finally, when sets get bigger than countable, combinatorial set theory offers some interesting ways other than cardinality to distinguish 'small' from 'large' sets. Among these are the use of clubs and stationary sets, of ultrafilters, and of measure and category. See P.Erdős, A.Hajnal, A. Máté & R. Rado, Combinatorial Set Theory: partition relations for cardinals, North-Holland 1984; W.W. Comfort & S. Negropontis, The Theory of Ultrafilters, Springer 1974; J.C. Oxtoby, Measure and Category: a survey of the analogies between topological and measure spaces (2nd ed.), Springer 1980. # Ramsey Theory for Graphs In this chapter we set out from a type of problem which, on the face of it, appears to be similar to the theme of the last chapter: what kind of substructures are necessarily present in every large enough graph? The regularity lemma of Chapter 7.4 provides one possible answer to this question: every (large) graph \( G \) contains large random-like subgraphs. If we are looking for a concrete interesting subgraph \( H \), on the other hand, our problem becomes more like Hadwiger's conjecture: we cannot expect an arbitrary graph \( G \) to contain a copy of \( H \), but if it does not then this might have some interesting structural implications for \( G \) . The kind of structural implication that will be typical for this chapter is simply that of containing some other (induced) subgraph. For example: given an integer \( r \), does every large enough graph contain either a \( {K}^{r} \) or an induced \( \overline{{K}^{r}\text{? }} \) Does every large enough connected graph contain either a \( {K}^{r} \) or else a large induced path or star? Despite its superficial similarity to extremal problems, the above type of question leads to a kind of mathematics with a distinctive flavour of its own. Indeed, the theorems and proofs in this chapter have more in common with similar results in algebra or geometry, say, than with most other areas of graph theory. The study of their underlying methods, therefore, is generally regarded as a combinatorial subject in its own right: the discipline of Ramsey theory. In line with the subject of this book, we shall focus on results that are naturally expressed in terms of graphs. Even from the viewpoint of general Ramsey theory, however, this is not as much of a limitation as it might seem: graphs are a natural setting for Ramsey problems, and the material in this chapter brings out a sufficient variety of ideas and methods to convey some of the fascination of the theory as a whole. ## 9.1 Ramsey's original theorems In its simplest version, Ramsey's theorem says that, given an integer \( r \geq 0 \), every large enough graph \( G \) contains either \( {K}^{r} \) or \( \overline{{K}^{r}} \) as an induced subgraph. At first glance, this may seem surprising: after all, we need about \( \left( {r - 2}\right) /\left( {r - 1}\right) \) of all possible edges to force a \( {K}^{r} \) subgraph in \( G \) (Corollary 7.1.3), but neither \( G \) nor \( \bar{G} \) can be expected to have more than half of all possible edges. However, as the Turán graphs illustrate well, squeezing many edges into \( G \) without creating a \( {K}^{r} \) imposes additional structure on \( G \), which may help us find an induced \( \overline{{K}^{r}} \) . So how could we go about proving Ramsey's theorem? Let us try to build a \( {K}^{r} \) or \( \overline{{K}^{r}} \) in \( G \) inductively, starting with an arbitrary vertex \( {v}_{1} \in {V}_{1} \mathrel{\text{:=}} V\left( G\right) \) . If \( \left| G\right| \) is large, there will be a large set \( {V}_{2} \subseteq {V}_{1} \smallsetminus \left\{ {v}_{1}\right\} \) of vertices that are either all adjacent to \( {v}_{1} \) or all non-adjacent to \( {v}_{1} \) . Accordingly, we may think of \( {v}_{1} \) as the first vertex of a \( {K}^{r} \) or \( \overline{{K}^{r}} \) whose other vertices all lie in \( {V}_{2} \) . Let us then choose another vertex \( {v}_{2} \in {V}_{2} \) for our \( {K}^{r} \) or \( \overline{{K}^{r}} \) . Since \( {V}_{2} \) is large, it will have a subset \( {V}_{3} \), still fairly large, of vertices that are all ’of the same type’ with respect to \( {v}_{2} \) as well: either all adjacent or all non-adjacent to it. We then continue our search for vertices inside \( {V}_{3} \), and so on (Fig. 9.1.1).  Fig. 9.1.1. Choosing the sequence \( {v}_{1},{v}_{2},\ldots \) How long can we go on in this way? This depends on the size of our initial set \( {V}_{1} \) : each set \( {V}_{i} \) has at least half the size of its predecessor \( {V}_{i - 1} \), so we shall be able to complete \( s \) construction steps if \( G \) has order about \( {2}^{s} \) . As the following proof shows, the choice of \( s = {2r} - 3 \) vertices \( {v}_{i} \) suffices to find among them the vertices of a \( {K}^{r} \) or \( \overline{{K}^{r}} \) . \( \left\lbrack {\;{9.2.2}\;}\right\rbrack \) Theorem 9.1.1. (Ramsey 1930) For every \( r \in \mathbb{N} \) there exists an \( n \in \mathbb{N} \) such that every graph of order at least \( n \) contains either \( {K}^{r} \) or \( \overline{{K}^{r}} \) as an induced subgraph. Proof. The assertion is trivial for \( r \leq 1 \) ; we assume that \( r \geq 2 \) . Let \( n \mathrel{\text{:=}} {2}^{{2r} - 3} \), and let \( G \) be a graph of order at least \( n \) . We shall define a sequence \( {V}_{1},\ldots ,{V}_{{2r} - 2} \) of sets and choose vertices \( {v}_{i} \in {V}_{i} \) with the following properties: (i) \( \left| {V}_{i}\right| = {2}^{{2r} - 2 - i}\;\left( {i = 1,\ldots ,{2r} - 2}\right) \) ; (ii) \( {V}_{i} \subseteq {V}_{i - 1} \smallsetminus \left\{ {v}_{i - 1}\right\} \;\left( {i = 2,\ldots ,{2r} - 2}\right) \) ; (iii) \( {v}_{i - 1} \) is adjacent either to all vertices in \( {V}_{i} \) or to no vertex in \( {V}_{i} \) \( \left( {i = 2,\ldots ,{2r} - 2}\right) \) . Let \( {V}_{1} \subseteq V\left( G\right) \) be any set of \( {2}^{{2r} - 3} \) vertices, and pick \( {v}_{1} \in {V}_{1} \) arbitrarily. Then (i) holds for \( i = 1 \), while (ii) and (iii) hold trivially. Suppose now that \( {V}_{i - 1} \) and \( {v}_{i - 1} \in {V}_{i - 1} \) have been chosen so as to satisfy (i)-(iii) for \( i - 1 \), where \( 1 < i \leq {2r} - 2 \) . Since \[ \left| {{V}_{i - 1} \smallsetminus \left\{ {v}_{i - 1}\right\} }\right| = {2}^{{2r} - 1 - i} - 1 \] is odd, \( {V}_{i - 1} \) has a subset \( {V}_{i} \) satisfying (i)-(iii); we pick \( {v}_{i} \in {V}_{i} \) arbitrarily. Among the \( {2r} - 3 \) vertices \( {v}_{1},\ldots ,{v}_{{2r} - 3} \), there are \( r - 1 \) vertices that show the same behaviour when viewed as \( {v}_{i - 1} \) in (iii), being adjacent either to all the vertices in \( {V}_{i} \) or to none. Accordingly, these \( r - 1 \) vertices and \( {v}_{{2r} - 2} \) induce either a \( {K}^{r} \) or a \( \overline{{K}^{r}} \) in \( G \), because \( {v}_{i},\ldots ,{v}_{{2r} - 2} \in {V}_{i} \) for all \( i \) . The least integer \( n \) associated with \( r \) as in Theorem 9.1.1 is the Ramsey --- Ramsey number \( R\left( r\right) \) --- number \( R\left( r\right) \) of \( r \) ; our proof shows that \( R\left( r\right) \leq {2}^{{2r} - 3} \) . In Chapter 11 we shall use a simple probabilistic argument to show that \( R\left( r\right) \) is bounded below by \( {2}^{r/2} \) (Theorem 11.1.3). It is customary in Ramsey theory to think of partitions as colourings: a colouring of (the elements of) a set \( X \) with \( c \) colours, or \( c \) -colouring for c-colouring short, is simply a partition of \( X \) into \( c \) classes (indexed by the ’colours’). In particular, these colourings need not satisfy any non-adjacency requirements as in Chapter 5. Given a \( c \) -colouring of \( {\left\lbrack X\right\rbrack }^{k} \), the set of all \( {\left\lbrack X\right\rbrack }^{k} \) \( k \) -subsets of \( X \), we call a set \( Y \subseteq X \) monochromatic if all the elements of \( {\left\lbrack Y\right\rbrack }^{k} \) have the same colour, \( {}^{1} \) i.e. belong to the same of the \( c \) partition mono- chromatic classes of \( {\left\lbrack X\right\rbrack }^{k} \) . Similarly, if \( G = \left( {V, E}\right) \) is a graph and all the edges of \( H \subseteq G \) have the same colour in some colouring of \( E \), we call \( H \) a monochromatic subgraph

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\) (Theorem 11.1.3). It is customary in Ramsey theory to think of partitions as colourings: a colouring of (the elements of) a set \( X \) with \( c \) colours, or \( c \) -colouring for c-colouring short, is simply a partition of \( X \) into \( c \) classes (indexed by the ’colours’). In particular, these colourings need not satisfy any non-adjacency requirements as in Chapter 5. Given a \( c \) -colouring of \( {\left\lbrack X\right\rbrack }^{k} \), the set of all \( {\left\lbrack X\right\rbrack }^{k} \) \( k \) -subsets of \( X \), we call a set \( Y \subseteq X \) monochromatic if all the elements of \( {\left\lbrack Y\right\rbrack }^{k} \) have the same colour, \( {}^{1} \) i.e. belong to the same of the \( c \) partition mono- chromatic classes of \( {\left\lbrack X\right\rbrack }^{k} \) . Similarly, if \( G = \left( {V, E}\right) \) is a graph and all the edges of \( H \subseteq G \) have the same colour in some colouring of \( E \), we call \( H \) a monochromatic subgraph of \( G \), speak of a red (green, etc.) \( H \) in \( G \), and so on. In the above terminology, Ramsey's theorem can be expressed as follows: for every \( r \) there exists an \( n \) such that, given any \( n \) -set \( X \) , every 2-colouring of \( {\left\lbrack X\right\rbrack }^{2} \) yields a monochromatic \( r \) -set \( Y \subseteq X \) . Interestingly, this assertion remains true for \( c \) -colourings of \( {\left\lbrack X\right\rbrack }^{k} \) with arbitrary \( c \) and \( k \) -with almost exactly the same proof! We first prove the infinite version, which is easier, and then deduce the finite version. Theorem 9.1.2. Let \( k, c \) be positive integers, and \( X \) an infinite set. If \( \left\lbrack {12.1.1}\right\rbrack \) \( {\left\lbrack X\right\rbrack }^{k} \) is coloured with \( c \) colours, then \( X \) has an infinite monochromatic subset. --- 1 Note that \( Y \) is called monochromatic, but it is the elements of \( {\left\lbrack Y\right\rbrack }^{k} \), not of \( Y \) , that are (equally) coloured. --- Proof. We prove the theorem by induction on \( k \), with \( c \) fixed. For \( k = 1 \) the assertion holds, so let \( k > 1 \) and assume the assertion for smaller values of \( k \) . Let \( {\left\lbrack X\right\rbrack }^{k} \) be coloured with \( c \) colours. We shall construct an infinite sequence \( {X}_{0},{X}_{1},\ldots \) of infinite subsets of \( X \) and choose elements \( {x}_{i} \in {X}_{i} \) with the following properties (for all \( i \) ): (i) \( {X}_{i + 1} \subseteq {X}_{i} \smallsetminus \left\{ {x}_{i}\right\} \) (ii) all \( k \) -sets \( \left\{ {x}_{i}\right\} \cup Z \) with \( Z \in {\left\lbrack {X}_{i + 1}\right\rbrack }^{k - 1} \) have the same colour, which we associate with \( {x}_{i} \) . We start with \( {X}_{0} \mathrel{\text{:=}} X \) and pick \( {x}_{0} \in {X}_{0} \) arbitrarily. By assumption, \( {X}_{0} \) is infinite. Having chosen an infinite set \( {X}_{i} \) and \( {x}_{i} \in {X}_{i} \) for some \( i \) , we \( c \) -colour \( {\left\lbrack {X}_{i} \smallsetminus \left\{ {x}_{i}\right\} \right\rbrack }^{k - 1} \) by giving each set \( Z \) the colour of \( \left\{ {x}_{i}\right\} \cup Z \) from our \( c \) -colouring of \( {\left\lbrack X\right\rbrack }^{k} \) . By the induction hypothesis, \( {X}_{i} \smallsetminus \left\{ {x}_{i}\right\} \) has an infinite monochromatic subset, which we choose as \( {X}_{i + 1} \) . Clearly, this choice satisfies (i) and (ii). Finally, we pick \( {x}_{i + 1} \in {X}_{i + 1} \) arbitrarily. Since \( c \) is finite, one of the \( c \) colours is associated with infinitely many \( {x}_{i} \) . These \( {x}_{i} \) form an infinite monochromatic subset of \( X \) . If desired, the finite version of Theorem 9.1.2 could be proved just like the infinite version above. However to ensure that the relevant sets are large enough at all stages of the induction, we have to keep track of their sizes, which involves a good deal of boring calculation. As long as we are not interested in bounds, the more elegant route is to deduce the finite version from the infinite 'by compactness', that is, using König's infinity lemma (8.1.2). \( \left\lbrack {9.3.3}\right\rbrack \) Theorem 9.1.3. For all \( k, c, r \geq 1 \) there exists an \( n \geq k \) such that every \( n \) -set \( X \) has a monochromatic \( r \) -subset with respect to any \( c \) -colouring of \( {\left\lbrack X\right\rbrack }^{k} \) . \( \left( {8.1.2}\right) \) Proof. As is customary in set theory, we denote by \( n \in \mathbb{N} \) (also) the \( k, c, r \) set \( \{ 0,\ldots, n - 1\} \) . Suppose the assertion fails for some \( k, c, r \) . Then for every \( n \geq k \) there exist an \( n \) -set, without loss of generality the set \( n \), and --- bad colouring --- a \( c \) -colouring \( {\left\lbrack n\right\rbrack }^{k} \rightarrow c \) such that \( n \) contains no monochromatic \( r \) -set. Let us call such colourings bad; we are thus assuming that for every \( n \geq k \) there exists a bad colouring of \( {\left\lbrack n\right\rbrack }^{k} \) . Our aim is to combine these into a bad colouring of \( {\left\lbrack \mathbb{N}\right\rbrack }^{k} \), which will contradict Theorem 9.1.2. For every \( n \geq k \) let \( {V}_{n} \neq \varnothing \) be the set of bad colourings of \( {\left\lbrack n\right\rbrack }^{k} \) . For \( n > k \), the restriction \( f\left( g\right) \) of any \( g \in {V}_{n} \) to \( {\left\lbrack n - 1\right\rbrack }^{k} \) is still bad, and hence lies in \( {V}_{n - 1} \) . By the infinity lemma (8.1.2), there is an infinite sequence \( {g}_{k},{g}_{k + 1},\ldots \) of bad colourings \( {g}_{n} \in {V}_{n} \) such that \( f\left( {g}_{n}\right) = {g}_{n - 1} \) for all \( n > k \) . For every \( m \geq k \), all colourings \( {g}_{n} \) with \( n \geq m \) agree on \( {\left\lbrack m\right\rbrack }^{k} \), so for each \( Y \in {\left\lbrack \mathbb{N}\right\rbrack }^{k} \) the value of \( {g}_{n}\left( Y\right) \) coincides for all \( n > \max Y \) . Let us define \( g\left( Y\right) \) as this common value \( {g}_{n}\left( Y\right) \) . Then \( g \) is a bad colouring of \( {\left\lbrack \mathbb{N}\right\rbrack }^{k} \) : every \( r \) -set \( S \subseteq \mathbb{N} \) is contained in some sufficiently large \( n \) , so \( S \) cannot be monochromatic since \( g \) coincides on \( {\left\lbrack n\right\rbrack }^{k} \) with the bad colouring \( {g}_{n} \) . The least integer \( n \) associated with \( k, c, r \) as in Theorem 9.1.3 is the --- Ramsey number \( R\left( {k, c, r}\right) \) --- Ramsey number for these parameters; we denote it by \( R\left( {k, c, r}\right) \) . ## 9.2 Ramsey numbers Ramsey’s theorem may be rephrased as follows: if \( H = {K}^{r} \) and \( G \) is a graph with sufficiently many vertices, then either \( G \) itself or its complement \( \bar{G} \) contains a copy of \( H \) as a subgraph. Clearly, the same is true for any graph \( H \), simply because \( H \subseteq {K}^{h} \) for \( h \mathrel{\text{:=}} \left| H\right| \) . However, if we ask for the least \( n \) such that every graph \( G \) of order \( n \) --- Ramsey number \( R\left( H\right) \) --- has the above property-this is the Ramsey number \( R\left( H\right) \) of \( H \) -then the above question makes sense: if \( H \) has only few edges, it should embed more easily in \( G \) or \( \bar{G} \), and we would expect \( R\left( H\right) \) to be smaller than the Ramsey number \( R\left( h\right) = R\left( {K}^{h}\right) \) . A little more generally, let \( R\left( {{H}_{1},{H}_{2}}\right) \) denote the least \( n \in \mathbb{N} \) such \( R\left( {{H}_{1},{H}_{2}}\right) \) that \( {H}_{1} \subseteq G \) or \( {H}_{2} \subseteq \bar{G} \) for every graph \( G \) of order \( n \) . For most graphs \( {H}_{1},{H}_{2} \), only very rough estimates are known for \( R\left( {{H}_{1},{H}_{2}}\right) \) . Interestingly, lower bounds given by random graphs (as in Theorem 11.1.3) are often sharper than even the best bounds provided by explicit constructions. The following proposition describes one of the few cases where exact Ramsey numbers are known for a relatively large class of graphs: Proposition 9.2.1. Let \( s, t \) be positive integers, and let \( T \) be a tree of order \( t \) . Then \( R\left( {T,{K}^{s}}\right) = \left( {s - 1}\right) \left( {t - 1}\right) + 1 \) . Proof. The disjoint union of \( s - 1 \) graphs \( {K}^{t - 1} \) contains no copy of \( T \) , \( \left( {5.2.3}\right) \) (1.5.4) while the complement of this graph, the complete \( \left( {s - 1}\right) \) -partite graph \( {K}_{t - 1}^{s - 1} \), does not contain \( {K}^{s} \) . This proves \( R\left( {T,{K}^{s}}\right) \geq \left( {s - 1}\right) \left( {t - 1}\right) + 1 \) . Conversely, let \( G \) be any graph of order \( n = \left( {s - 1}\right) \left( {t - 1}\right) + 1 \) whose complement contains no \( {K}^{s} \) . Then \( s > 1 \), and in any vertex colouring of \( G \) (in the sense of Chapter 5) at most \( s - 1 \) vertices can have the same colour. Hence, \( \chi \left( G\right) \geq \lceil n/\left( {s - 1}\right) \rceil = t \) . By Corollary 5.2.3, \( G \) has a subgraph \( H \) with \( \delta \left( H\right) \geq t - 1 \), which by Corollary 1.5.4 contains a copy of \( T \) . As the main result of this section, we shall now prove one of those rare general theorems providing a relatively good upper bound for the Ramsey numbers of a large class of graphs, a class defined in terms of a standard graph invariant. The theorem deals with the Ramsey numbers of sparse graphs: it says that the Ramsey number of graphs \( H \) with bounded maximum degree grows only linearly in \( \left| H\right| \) -an enormous improvement on the exponential bound from the proof of Theorem 9.1.1. Theorem 9.2.2. (Chvátal, Rödl, Szemerédi & Trotter 1983) For every positive integer \( \Delta \) there is a constant \( c \) such that \[ R\left( H\right) \leq c\le

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g of \( G \) (in the sense of Chapter 5) at most \( s - 1 \) vertices can have the same colour. Hence, \( \chi \left( G\right) \geq \lceil n/\left( {s - 1}\right) \rceil = t \) . By Corollary 5.2.3, \( G \) has a subgraph \( H \) with \( \delta \left( H\right) \geq t - 1 \), which by Corollary 1.5.4 contains a copy of \( T \) . As the main result of this section, we shall now prove one of those rare general theorems providing a relatively good upper bound for the Ramsey numbers of a large class of graphs, a class defined in terms of a standard graph invariant. The theorem deals with the Ramsey numbers of sparse graphs: it says that the Ramsey number of graphs \( H \) with bounded maximum degree grows only linearly in \( \left| H\right| \) -an enormous improvement on the exponential bound from the proof of Theorem 9.1.1. Theorem 9.2.2. (Chvátal, Rödl, Szemerédi & Trotter 1983) For every positive integer \( \Delta \) there is a constant \( c \) such that \[ R\left( H\right) \leq c\left| H\right| \] for all graphs \( H \) with \( \Delta \left( H\right) \leq \Delta \) . Proof. The basic idea of the proof is as follows. We wish to show that \( H \subseteq G \) or \( H \subseteq \bar{G} \) if \( \left| G\right| \) is large enough (though not too large). Consider an \( \epsilon \) -regular partition of \( G \), as provided by the regularity lemma. If enough of the \( \epsilon \) -regular pairs in this partition have high density, we may hope to find a copy of \( H \) in \( G \) . If most pairs have low density, we try to find \( H \) in \( \bar{G} \) . Let \( R,{R}^{\prime } \) and \( {R}^{\prime \prime } \) be the regularity graphs of \( G \) whose edges correspond to the pairs of density \( \geq 0; \geq 1/2; < 1/2 \) respectively. \( {}^{2} \) Then \( R \) is the edge-disjoint union of \( {R}^{\prime } \) and \( {R}^{\prime \prime } \) . Now to obtain \( H \subseteq G \) or \( H \subseteq \bar{G} \), it suffices by Lemma 7.5.2 to ensure that \( H \) is contained in a suitable ’inflated regularity graph’ \( {R}_{s}^{\prime } \) or \( {R}_{s}^{\prime \prime } \) . Since \( \chi \left( H\right) \leq \Delta \left( H\right) + 1 \leq \Delta + 1 \), this will be the case if \( s \geq \alpha \left( H\right) \) and we can find a \( {K}^{\Delta + 1} \) in \( {R}^{\prime } \) or in \( {R}^{\prime \prime } \) . But that is easy to ensure: we just need that \( {K}^{r} \subseteq R \), where \( r \) is the Ramsey number of \( \Delta + 1 \), which will follow from Turán’s theorem because \( R \) is dense. \( \Delta, d \) For the formal proof let now \( \Delta \geq 1 \) be given. On input \( d \mathrel{\text{:=}} 1/2 \) \( {\epsilon }_{0}, m \) and \( \Delta \), Lemma 7.5.2 returns an \( {\epsilon }_{0} \) . Let \( m \mathrel{\text{:=}} R\left( {\Delta + 1}\right) \) be the Ramsey \( \epsilon \) number of \( \Delta + 1 \) . Let \( \epsilon \leq {\epsilon }_{0} \) be positive but small enough that for \( k = m \) (and hence for all \( k \geq m \) ) \[ {2\epsilon } < \frac{1}{m - 1} - \frac{1}{k} \] (1) \( M \) then in particular \( \epsilon < 1 \) . Finally, let \( M \) be the integer returned by the regularity lemma (7.4.1) on input \( \epsilon \) and \( m \) . All the quantities defined so far depend only on \( \Delta \) . We shall prove the theorem with \( c \) \[ c \mathrel{\text{:=}} \frac{{2}^{\Delta + 1}M}{1 - \epsilon }. \] Let \( H \) with \( \Delta \left( H\right) \leq \Delta \) be given, and let \( s \mathrel{\text{:=}} \left| H\right| \) . Let \( G \) be an arbitrary \( G, n \) graph of order \( n \geq c\left| H\right| \) ; we show that \( H \subseteq G \) or \( H \subseteq \bar{G} \) . 2 In our formal proof later we shall define \( {R}^{\prime \prime } \) a little differently, so that it complies properly with our definition of a regularity graph. By Lemma 7.4.1, \( G \) has an \( \epsilon \) -regular partition \( \left\{ {{V}_{0},{V}_{1},\ldots ,{V}_{k}}\right\} \) with exceptional set \( {V}_{0} \) and \( \left| {V}_{1}\right| = \ldots = \left| {V}_{k}\right| = : \ell \), where \( m \leq k \leq M \) . Then \[ \ell = \frac{n - \left| {V}_{0}\right| }{k} \geq n\frac{1 - \epsilon }{M} \geq {cs}\frac{1 - \epsilon }{M} \geq {2}^{\Delta + 1}s = {2s}/{d}^{\Delta }. \] (2) Let \( R \) be the regularity graph with parameters \( \epsilon ,\ell ,0 \) corresponding to this partition. By definition, \( R \) has \( k \) vertices and \[ \parallel R\parallel \geq \left( \begin{array}{l} k \\ 2 \end{array}\right) - \epsilon {k}^{2} \] \[ = \frac{1}{2}{k}^{2}\left( {1 - \frac{1}{k} - {2\epsilon }}\right) \] \[ \underset{\left( 1\right) }{ > }\frac{1}{2}{k}^{2}\left( {1 - \frac{1}{k} - \frac{1}{m - 1} + \frac{1}{k}}\right) \] \[ = \frac{1}{2}{k}^{2}\frac{m - 2}{m - 1} \] \[ \geq {t}_{m - 1}\left( k\right) \] edges. By Theorem 7.1.1, therefore, \( R \) has a subgraph \( K = {K}^{m} \) . We now colour the edges of \( R \) with two colours: red if the edge corresponds to a pair \( \left( {{V}_{i},{V}_{j}}\right) \) of density at least \( 1/2 \), and green otherwise. Let \( {R}^{\prime } \) be the spanning subgraph of \( R \) formed by the red edges, and \( {R}^{\prime \prime } \) the spanning subgraph of \( R \) formed by the green edges and those whose corresponding pair has density exactly \( 1/2 \) . Then \( {R}^{\prime } \) is a regularity graph of \( G \) with parameters \( \epsilon ,\ell \) and \( 1/2 \) . And \( {R}^{\prime \prime } \) is a regularity graph of \( \bar{G} \) , with the same parameters: as one easily checks, every pair \( \left( {{V}_{i},{V}_{j}}\right) \) that is \( \epsilon \) -regular for \( G \) is also \( \epsilon \) -regular for \( \bar{G} \) . By definition of \( m \), our graph \( K \) contains a red or a green \( {K}^{r} \), for \( r \mathrel{\text{:=}} \chi \left( H\right) \leq \Delta + 1 \) . Correspondingly, \( H \subseteq {R}_{s}^{\prime } \) or \( H \subseteq {R}_{s}^{\prime \prime } \) . Since \( \epsilon \leq {\epsilon }_{0} \) and \( \ell \geq {2s}/{d}^{\Delta } \) by (2), both \( {R}^{\prime } \) and \( {R}^{\prime \prime } \) satisfy the requirements of Lemma 7.5.2, so \( H \subseteq G \) or \( H \subseteq \bar{G} \) as desired. So far in this section, we have been asking what is the least order of a graph \( G \) such that every 2-colouring of its edges yields a monochromatic copy of some given graph \( H \) . Rather than focusing on the order of \( G \), we might alternatively try to minimize \( G \) itself, with respect to the subgraph --- Ramsey-minimal --- relation. Given a graph \( H \), let us call a graph \( G \) Ramsey-minimal for \( H \) if \( G \) is minimal with the property that every 2-colouring of its edges yields a monochromatic copy of \( H \) . What do such Ramsey-minimal graphs look like? Are they unique? The following result, which we include for its pretty proof, answers the second question for some \( H \) : Proposition 9.2.3. If \( T \) is a tree but not a star, then infinitely many graphs are Ramsey-minimal for \( T \) . Proof. Let \( \left| T\right| = : r \) . We show that for every \( n \in \mathbb{N} \) there is a graph of order at least \( n \) that is Ramsey-minimal for \( T \) . By Theorem 5.2.5, there exists a graph \( G \) with chromatic number \( \chi \left( G\right) > {r}^{2} \) and girth \( g\left( G\right) > n \) . If we colour the edges of \( G \) red and green, then the red and the green subgraph cannot both have an \( r \) - (vertex-)colouring in the sense of Chapter 5: otherwise we could colour the vertices of \( G \) with the pairs of colours from those colourings and obtain a contradiction to \( \chi \left( G\right) > {r}^{2} \) . So let \( {G}^{\prime } \subseteq G \) be monochromatic with \( \chi \left( {G}^{\prime }\right) > r \) . By Corollary 5.2.3, \( {G}^{\prime } \) has a subgraph of minimum degree at least \( r \), which contains a copy of \( T \) by Corollary 1.5.4. Let \( {G}^{ * } \subseteq G \) be Ramsey-minimal for \( T \) . Clearly, \( {G}^{ * } \) is not a forest: the edges of any forest can be 2-coloured (partitioned) so that no monochromatic subforest contains a path of length 3 , let alone a copy of \( T \) . (Here we use that \( T \) is not a star, and hence contains a \( {P}^{3} \) .) So \( {G}^{ * } \) contains a cycle, which has length \( g\left( G\right) > n \) since \( {G}^{ * } \subseteq G \) . In particular, \( \left| {G}^{ * }\right| > n \) as desired. ## 9.3 Induced Ramsey theorems Ramsey’s theorem can be rephrased as follows. For every graph \( H = {K}^{r} \) there exists a graph \( G \) such that every 2-colouring of the edges of \( G \) yields a monochromatic \( H \subseteq G \) ; as it turns out, this is witnessed by any large enough complete graph as \( G \) . Let us now change the problem slightly and ask for a graph \( G \) in which every 2-edge-colouring yields a monochromatic induced \( H \subseteq G \), where \( H \) is now an arbitrary given graph. This slight modification changes the character of the problem dramatically. What is needed now is no longer a simple proof that \( G \) is 'big enough' (as for Theorem 9.1.1), but a careful construction: the construction of a graph that, however we bipartition its edges, contains --- Ramsey graph --- an induced copy of \( H \) with all edges in one partition class. We shall call such a graph a Ramsey graph for \( H \) . The fact that such a Ramsey graph exists for every choice of \( H \) is one of the fundamental results of graph Ramsey theory. It was proved around 1973, independently by Deuber, by Erdős, Hajnal & Pósa, and by Rödl. Theorem 9.3.1. Every graph has a Ramsey graph. In other words, for every graph \( H \) there exists a graph \( G \) that, for every partition \( \left\{ {{E}_{1},{E}_{2}}\right\} \) of \( E\left( G\right) \), has an induced subgraph \( H \) with \( E\left( H\right) \subseteq {E}_{1} \) or \( E\left( H\right) \subseteq {E}_{2} \) . We give two proofs. Each of these is highly individual, yet each off

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the problem dramatically. What is needed now is no longer a simple proof that \( G \) is 'big enough' (as for Theorem 9.1.1), but a careful construction: the construction of a graph that, however we bipartition its edges, contains --- Ramsey graph --- an induced copy of \( H \) with all edges in one partition class. We shall call such a graph a Ramsey graph for \( H \) . The fact that such a Ramsey graph exists for every choice of \( H \) is one of the fundamental results of graph Ramsey theory. It was proved around 1973, independently by Deuber, by Erdős, Hajnal & Pósa, and by Rödl. Theorem 9.3.1. Every graph has a Ramsey graph. In other words, for every graph \( H \) there exists a graph \( G \) that, for every partition \( \left\{ {{E}_{1},{E}_{2}}\right\} \) of \( E\left( G\right) \), has an induced subgraph \( H \) with \( E\left( H\right) \subseteq {E}_{1} \) or \( E\left( H\right) \subseteq {E}_{2} \) . We give two proofs. Each of these is highly individual, yet each offers a glimpse of true Ramsey theory: the graphs involved are used as hardly more than bricks in the construction, but the edifice is impressive. First proof. In our construction of the desired Ramsey graph we shall repeatedly replace vertices of a graph \( G = \left( {V, E}\right) \) already constructed by copies of another graph \( H \) . For a vertex set \( U \subseteq V \) let \( G\left\lbrack {U \rightarrow H}\right\rbrack \) \( G\left\lbrack {U \rightarrow H}\right\rbrack \) denote the graph obtained from \( G \) by replacing the vertices \( u \in U \) with copies \( H\left( u\right) \) of \( H \) and joining each \( H\left( u\right) \) completely to all \( H\left( {u}^{\prime }\right) \) with \( H\left( u\right) \) \( u{u}^{\prime } \in E \) and to all vertices \( v \in V \smallsetminus U \) with \( {uv} \in E \) (Fig. 9.3.1). Formally,  Fig. 9.3.1. A graph \( G\left\lbrack {U \rightarrow H}\right\rbrack \) with \( H = {K}^{3} \) \( G\left\lbrack {U \rightarrow H}\right\rbrack \) is the graph on \[ \left( {U \times V\left( H\right) }\right) \cup \left( {\left( {V \smallsetminus U}\right) \times \{ \varnothing \} }\right) \] in which two vertices \( \left( {v, w}\right) \) and \( \left( {{v}^{\prime },{w}^{\prime }}\right) \) are adjacent if and only if either \( v{v}^{\prime } \in E \), or else \( v = {v}^{\prime } \in U \) and \( w{w}^{\prime } \in E\left( H\right) {.}^{3} \) We prove the following formal strengthening of Theorem 9.3.1: For any two graphs \( {H}_{1},{H}_{2} \) there exists a graph \( G = \) \( G\left( {{H}_{1},{H}_{2}}\right) \) such that every edge colouring of \( G \) with the \( G\left( {{H}_{1},{H}_{2}}\right) \) colours 1 and 2 yields either an induced \( {H}_{1} \subseteq G \) with all \( \left( *\right) \) its edges coloured 1 or an induced \( {H}_{2} \subseteq G \) with all its edges coloured 2. This formal strengthening makes it possible to apply induction on \( \left| {H}_{1}\right| + \left| {H}_{2}\right| \), as follows. If either \( {H}_{1} \) or \( {H}_{2} \) has no edges (in particular, if \( \left| {H}_{1}\right| + \left| {H}_{2}\right| \leq 1 \) ), then (*) holds with \( G = \overline{{K}^{n}} \) for large enough \( n \) . For the induction step, we now assume that both \( {H}_{1} \) and \( {H}_{2} \) have at least one edge, and that \( \left( *\right) \) holds for all pairs \( \left( {{H}_{1}^{\prime },{H}_{2}^{\prime }}\right) \) with smaller \( \left| {H}_{1}^{\prime }\right| + \left| {H}_{2}^{\prime }\right| \) . --- 3 The replacement of \( V \smallsetminus U \) by \( \left( {V \smallsetminus U}\right) \times \{ \varnothing \} \) is just a formal device to ensure that all vertices of \( G\left\lbrack {U \rightarrow H}\right\rbrack \) have the same form \( \left( {v, w}\right) \), and that \( G\left\lbrack {U \rightarrow H}\right\rbrack \) is formally disjoint from \( G \) . --- \( {x}_{i} \) For each \( i = 1,2 \), pick a vertex \( {x}_{i} \in {H}_{i} \) that is incident with an \( {H}_{i}^{\prime },{H}_{i}^{\prime \prime } \) edge. Let \( {H}_{i}^{\prime } \mathrel{\text{:=}} {H}_{i} - {x}_{i} \), and let \( {H}_{i}^{\prime \prime } \) be the subgraph of \( {H}_{i}^{\prime } \) induced by the neighbours of \( {x}_{i} \) . We shall construct a sequence \( {G}^{0},\ldots ,{G}^{n} \) of disjoint graphs; \( {G}^{n} \) will be the desired Ramsey graph \( G\left( {{H}_{1},{H}_{2}}\right) \) . Along with the graphs \( {G}_{i} \), we shall define subsets \( {V}^{i} \subseteq V\left( {G}^{i}\right) \) and a map \[ f : {V}^{1} \cup \ldots \cup {V}^{n} \rightarrow {V}^{0} \cup \ldots \cup {V}^{n - 1} \] such that \[ f\left( {V}^{i}\right) = {V}^{i - 1} \] (1) \( {f}^{i} \) for all \( i \geq 1 \) . Writing \( {f}^{i} \mathrel{\text{:=}} f \circ \ldots \circ f \) for the \( i \) -fold composition of \( f \) , and \( {f}^{0} \) for the identity map on \( {V}^{0} = V\left( {G}^{0}\right) \), we thus have \( {f}^{i}\left( v\right) \in {V}^{0} \) origin for all \( v \in {V}^{i} \) . We call \( {f}^{i}\left( v\right) \) the origin of \( v \) . The subgraphs \( {G}^{i}\left\lbrack {V}^{i}\right\rbrack \) will reflect the structure of \( {G}^{0} \) as follows: Vertices in \( {V}^{i} \) with different origins are adjacent in \( {G}^{i} \) if (2) and only if their origins are adjacent in \( {G}^{0} \) . Assertion (2) will not be used formally in the proof below. However, it can help us to visualize the graphs \( {G}^{i} \) : every \( {G}^{i} \) (more precisely, every \( {G}^{i}\left\lbrack {V}^{i}\right\rbrack \) -there will also be some vertices \( x \in {G}^{i} - {V}^{i} \) ) is essentially an inflated copy of \( {G}^{0} \) in which every vertex \( w \in {G}^{0} \) has been replaced by the set of all vertices in \( {V}^{i} \) with origin \( w \), and the map \( f \) links vertices with the same origin across the various \( {G}^{i} \) . By the induction hypothesis, there are Ramsey graphs \( {G}_{1},{G}_{2} \) \[ {G}_{1} \mathrel{\text{:=}} G\left( {{H}_{1},{H}_{2}^{\prime }}\right) \;\text{ and }\;{G}_{2} \mathrel{\text{:=}} G\left( {{H}_{1}^{\prime },{H}_{2}}\right) . \] \( {G}^{0},{V}^{0} \) Let \( {G}^{0} \) be a copy of \( {G}_{1} \), and set \( {V}^{0} \mathrel{\text{:=}} V\left( {G}^{0}\right) \) . Let \( {W}_{0}^{\prime },\ldots ,{W}_{n - 1}^{\prime } \) be the \( {W}_{i}^{\prime } \) subsets of \( {V}^{0} \) spanning an \( {H}_{2}^{\prime } \) in \( {G}^{0} \) . Thus, \( n \) is defined as the number \( n \) of induced copies of \( {H}_{2}^{\prime } \) in \( {G}^{0} \), and we shall construct a graph \( {G}^{i} \) for \( {W}_{i}^{\prime \prime } \) every set \( {W}_{i - 1}^{\prime }, i = 1,\ldots, n \) . For \( i = 0,\ldots, n - 1 \), let \( {W}_{i}^{\prime \prime } \) be the image of \( V\left( {H}_{2}^{\prime \prime }\right) \) under some isomorphism \( {H}_{2}^{\prime } \rightarrow {G}^{0}\left\lbrack {W}_{i}^{\prime }\right\rbrack \) . Assume now that \( {G}^{0},\ldots ,{G}^{i - 1} \) and \( {V}^{0},\ldots ,{V}^{i - 1} \) have been defined for some \( i \geq 1 \), and that \( f \) has been defined on \( {V}^{1} \cup \ldots \cup {V}^{i - 1} \) and satisfies (1) for all \( j \leq i \) . We construct \( {G}^{i} \) from \( {G}^{i - 1} \) in two steps. For \( {U}^{i - 1} \) the first step, consider the set \( {U}^{i - 1} \) of all the vertices \( v \in {V}^{i - 1} \) whose origin \( {f}^{i - 1}\left( v\right) \) lies in \( {W}_{i - 1}^{\prime \prime } \) . (For \( i = 1 \), this gives \( {U}^{0} = {W}_{0}^{\prime \prime } \) .) Expand \( {G}^{i - 1} \) to a new graph \( {\widetilde{G}}^{i - 1} \) (disjoint from \( {G}^{i - 1} \) ) by replacing every vertex \( {G}_{2}\left( u\right) \) \( u \in {U}^{i - 1} \) with a copy \( {G}_{2}\left( u\right) \) of \( {G}_{2} \), i.e. let \( {\widetilde{G}}^{i - 1} \) \[ {\widetilde{G}}^{i - 1} \mathrel{\text{:=}} {G}^{i - 1}\left\lbrack {{U}^{i - 1} \rightarrow {G}_{2}}\right\rbrack \]  Fig. 9.3.2. The construction of \( {G}^{1} \) (see Figures 9.3.2 and 9.3.3). Set \( f\left( {u}^{\prime }\right) \mathrel{\text{:=}} u \) for all \( u \in {U}^{i - 1} \) and \( {u}^{\prime } \in {G}_{2}\left( u\right) \), and \( f\left( {v}^{\prime }\right) \mathrel{\text{:=}} v \) for all \( {v}^{\prime } = \left( {v,\varnothing }\right) \) with \( v \in {V}^{i - 1} \smallsetminus {U}^{i - 1} \) . (Recall that \( \left( {v,\varnothing }\right) \) is simply the unexpanded copy of a vertex \( v \in {G}^{i - 1} \) in \( {\widetilde{G}}^{i - 1} \) .) Let \( {V}^{i} \) be the set of those vertices \( {v}^{\prime } \) or \( {u}^{\prime } \) of \( {\widetilde{G}}^{i - 1} \) for which \( f \) has thus been defined, i.e. the vertices that either correspond directly to a vertex \( v \) in \( {V}^{i - 1} \) or else belong to an expansion \( {G}_{2}\left( u\right) \) of such a vertex \( u \) . Then (1) holds for \( i \) . Also, if we assume (2) inductively for \( i - 1 \), then (2) holds again for \( i \) (in \( {\widetilde{G}}^{i - 1} \) ). The graph \( {\widetilde{G}}^{i - 1} \) is already the essential part of \( {G}^{i} \) : the part that looks like an inflated copy of \( {G}^{0} \) . In the second step we now extend \( {\widetilde{G}}^{i - 1} \) to the desired graph \( {G}^{i} \) by adding some further vertices \( x \notin {V}^{i} \) . Let \( \mathcal{F} \) denote the set of all families \( F \) of the form \[ F = \left( {{H}_{1}^{\prime }\left( u\right) \mid u \in {U}^{i - 1}}\right) \] where each \( {H}_{1}^{\prime }\left( u\right) \) is an induced subgraph of \( {G}_{2}\left( u\right) \) isomorphic to \( {H}_{1}^{\prime } \) . \( {H}_{1}^{\prime }\left( u\right) \) (Less formally: \( \mathcal{F} \) is the collection of ways to select simultaneously from each \( {G}_{2}\left( u\right) \) exactly one induced copy of \( {H}_

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( u\right) \) of such a vertex \( u \) . Then (1) holds for \( i \) . Also, if we assume (2) inductively for \( i - 1 \), then (2) holds again for \( i \) (in \( {\widetilde{G}}^{i - 1} \) ). The graph \( {\widetilde{G}}^{i - 1} \) is already the essential part of \( {G}^{i} \) : the part that looks like an inflated copy of \( {G}^{0} \) . In the second step we now extend \( {\widetilde{G}}^{i - 1} \) to the desired graph \( {G}^{i} \) by adding some further vertices \( x \notin {V}^{i} \) . Let \( \mathcal{F} \) denote the set of all families \( F \) of the form \[ F = \left( {{H}_{1}^{\prime }\left( u\right) \mid u \in {U}^{i - 1}}\right) \] where each \( {H}_{1}^{\prime }\left( u\right) \) is an induced subgraph of \( {G}_{2}\left( u\right) \) isomorphic to \( {H}_{1}^{\prime } \) . \( {H}_{1}^{\prime }\left( u\right) \) (Less formally: \( \mathcal{F} \) is the collection of ways to select simultaneously from each \( {G}_{2}\left( u\right) \) exactly one induced copy of \( {H}_{1}^{\prime } \) .) For each \( F \in \mathcal{F} \), add a vertex \( x\left( F\right) \) to \( {\widetilde{G}}^{i - 1} \) and join it, for every \( u \in {U}^{i - 1} \), to all the vertices in \( x\left( F\right) \) the image \( {H}_{1}^{\prime \prime }\left( u\right) \subseteq {H}_{1}^{\prime }\left( u\right) \) of \( {H}_{1}^{\prime \prime } \) under some isomorphism from \( {H}_{1}^{\prime } \) to \( {H}_{1}^{\prime \prime }\left( u\right) \) the \( {H}_{1}^{\prime }\left( u\right) \subseteq {G}_{2}\left( u\right) \) selected by \( F \) (Fig. 9.3.2). Denote the resulting graph by \( {G}^{i} \) . This completes the inductive definition of the graphs \( {G}^{0},\ldots ,{G}^{n} \) . Let us now show that \( G \mathrel{\text{:=}} {G}^{n} \) satisfies \( \left( *\right) \) . To this end, we prove the following assertion \( \left( {* * }\right) \) about \( {G}^{i} \) for \( i = 0,\ldots, n \) : For every edge colouring with the colours 1 and 2, \( {G}^{i} \) contains either an induced \( {H}_{1} \) coloured 1, or an induced \( {H}_{2} \) coloured 2, or an induced subgraph \( H \) coloured 2 such that \( \left( {* * }\right) \) \( V\left( H\right) \subseteq {V}^{i} \) and the restriction of \( {f}^{i} \) to \( V\left( H\right) \) is an isomorphism between \( H \) and \( {G}^{0}\left\lbrack {W}_{k}^{\prime }\right\rbrack \) for some \( k \in \{ i,\ldots, n - 1\} \) . Note that the third of the above cases cannot arise for \( i = n \), so \( \left( {* * }\right) \) for \( n \) is equivalent to \( \left( *\right) \) with \( G \mathrel{\text{:=}} {G}^{n} \) . For \( i = 0,\left( {* * }\right) \) follows from the choice of \( {G}^{0} \) as a copy of \( {G}_{1} = \) \( G\left( {{H}_{1},{H}_{2}^{\prime }}\right) \) and the definition of the sets \( {W}_{k}^{\prime } \) . Now let \( 1 \leq i \leq n \), and assume \( \left( {* * }\right) \) for smaller values of \( i \) . Let an edge colouring of \( {G}^{i} \) be given. For each \( u \in {U}^{i - 1} \) there is a copy of \( {G}_{2} \) in \( {G}^{i} \) : \[ {G}^{i} \supseteq {G}_{2}\left( u\right) \simeq G\left( {{H}_{1}^{\prime },{H}_{2}}\right) \] If \( {G}_{2}\left( u\right) \) contains an induced \( {H}_{2} \) coloured 2 for some \( u \in {U}^{i - 1} \), we are done. If not, then every \( {G}_{2}\left( u\right) \) has an induced subgraph \( {H}_{1}^{\prime }\left( u\right) \simeq {H}_{1}^{\prime } \) coloured 1. Let \( F \) be the family of these graphs \( {H}_{1}^{\prime }\left( u\right) \), one for each \( u \in {U}^{i - 1} \), and let \( x \mathrel{\text{:=}} x\left( F\right) \) . If, for some \( u \in {U}^{i - 1} \), all the \( x - {H}_{1}^{\prime \prime }\left( u\right) \) edges in \( {G}^{i} \) are also coloured 1, we have an induced copy of \( {H}_{1} \) in \( {G}^{i} \) and are again done. We may therefore assume that each \( {H}_{1}^{\prime \prime }\left( u\right) \) has a \( {y}_{u} \) vertex \( {y}_{u} \) for which the edge \( x{y}_{u} \) is coloured 2 . The restriction \( {y}_{u} \mapsto u \) of \( f \) to \( {\widehat{U}}^{i - 1} \) \[ {\widehat{U}}^{i - 1} \mathrel{\text{:=}} \left\{ {{y}_{u} \mid u \in {U}^{i - 1}}\right\} \subseteq {V}^{i} \] extends by \( \left( {v,\varnothing }\right) \mapsto v \) to an isomorphism from \( {\widehat{G}}^{i - 1} \) \[ {\widehat{G}}^{i - 1} \mathrel{\text{:=}} {G}^{i}\left\lbrack {{\widehat{U}}^{i - 1} \cup \left\{ {\left( {v,\varnothing }\right) \mid v \in V\left( {G}^{i - 1}\right) \smallsetminus {U}^{i - 1}}\right\} }\right\rbrack \] to \( {G}^{i - 1} \), and so our edge colouring of \( {G}^{i} \) induces an edge colouring of \( {G}^{i - 1} \) . If this colouring yields an induced \( {H}_{1} \subseteq {G}^{i - 1} \) coloured 1 or an induced \( {H}_{2} \subseteq {G}^{i - 1} \) coloured 2, we have these also in \( {\widehat{G}}^{i - 1} \subseteq {G}^{i} \) and are again home. By \( \left( {* * }\right) \) for \( i - 1 \) we may therefore assume that \( {G}^{i - 1} \) has an induced subgraph \( {H}^{\prime } \) coloured 2, with \( V\left( {H}^{\prime }\right) \subseteq {V}^{i - 1} \), and such that the restriction of \( {f}^{i - 1} \) to \( V\left( {H}^{\prime }\right) \) is an isomorphism from \( {H}^{\prime } \) to \( {G}^{0}\left\lbrack {W}_{k}^{\prime }\right\rbrack \simeq {H}_{2}^{\prime } \) for some \( k \in \{ i - 1,\ldots, n - 1\} \) . Let \( {\widehat{H}}^{\prime } \) be the corresponding induced subgraph of \( {\widehat{G}}^{i - 1} \subseteq {G}^{i} \) (also coloured 2); then \( V\left( {\widehat{H}}^{\prime }\right) \subseteq {V}^{i} \) , \[ {f}^{i}\left( {V\left( {\widehat{H}}^{\prime }\right) }\right) = {f}^{i - 1}\left( {V\left( {H}^{\prime }\right) }\right) = {W}_{k}^{\prime }, \] and \( {f}^{i} : {\widehat{H}}^{\prime } \rightarrow {G}^{0}\left\lbrack {W}_{k}^{\prime }\right\rbrack \) is an isomorphism. If \( k \geq i \), this completes the proof of \( \left( {* * }\right) \) with \( H \mathrel{\text{:=}} {\widehat{H}}^{\prime } \) ; we therefore assume that \( k < i \), and hence \( k = i - 1 \) (Fig. 9.3.3). By definition of \( {U}^{i - 1} \) and \( {\widehat{G}}^{i - 1} \), the inverse image of \( {W}_{i - 1}^{\prime \prime } \) under the isomorphism \( {f}^{i} : {\widehat{H}}^{\prime } \rightarrow {G}^{0}\left\lbrack {W}_{i - 1}^{\prime }\right\rbrack \) is a subset of \( {\widetilde{U}}^{i - 1} \) . Since \( x \) is joined to precisely those vertices of \( {\widehat{H}}^{\prime } \) that lie in \( {\widehat{U}}^{i - 1} \), and all these edges \( x{y}_{u} \) have colour 2, the graph \( {\widehat{H}}^{\prime } \) and \( x \) together induce in \( {G}^{i} \) a copy of \( {H}_{2} \) coloured 2, and the proof of \( \left( {* * }\right) \) is complete.  Fig. 9.3.3. A monochromatic copy of \( {H}_{2} \) in \( {G}^{i} \) Let us return once more to the reformulation of Ramsey's theorem considered at the beginning of this section: for every graph \( H \) there exists a graph \( G \) such that every 2-colouring of the edges of \( G \) yields a monochromatic \( H \subseteq G \) . The graph \( G \) for which this follows at once from Ramsey's theorem is a sufficiently large complete graph. If we ask, however, that \( G \) shall not contain any complete subgraphs larger than those in \( H \), i.e. that \( \omega \left( G\right) = \omega \left( H\right) \), the problem again becomes difficult - even if we do not require \( H \) to be induced in \( G \) . Our second proof of Theorem 9.3.1 solves both problems at once: given \( H \), we shall construct a Ramsey graph for \( H \) with the same clique number as \( H \) . For this proof, i.e. for the remainder of this section, let us view bipartite graphs \( P \) as triples \( \left( {{V}_{1},{V}_{2}, E}\right) \), where \( {V}_{1} \) and \( {V}_{2} \) are the two bipartite vertex classes and \( E \subseteq {V}_{1} \times {V}_{2} \) is the set of edges. The reason for this more explicit notation is that we want embeddings between bipartite graphs to respect their bipartitions: given another bipartite graph \( {P}^{\prime } = \) \( \left( {{V}_{1}^{\prime },{V}_{2}^{\prime },{E}^{\prime }}\right) \), an injective map \( \varphi : {V}_{1} \cup {V}_{2} \rightarrow {V}_{1}^{\prime } \cup {V}_{2}^{\prime } \) will be called an embedding of \( P \) in \( {P}^{\prime } \) if \( \varphi \left( {V}_{i}\right) \subseteq {V}_{i}^{\prime } \) for \( i = 1,2 \) and \( \varphi \left( {v}_{1}\right) \varphi \left( {v}_{2}\right) \) is an edge --- embedding \( P \rightarrow {P}^{\prime } \) --- of \( {P}^{\prime } \) if and only if \( {v}_{1}{v}_{2} \) is an edge of \( P \) . (Note that such embeddings are ’induced’.) Instead of \( \varphi : {V}_{1} \cup {V}_{2} \rightarrow {V}_{1}^{\prime } \cup {V}_{2}^{\prime } \) we may simply write \( \varphi : P \rightarrow {P}^{\prime } \) . We need two lemmas. Lemma 9.3.2. Every bipartite graph can be embedded in a bipartite graph of the form \( \left( {X,{\left\lbrack X\right\rbrack }^{k}, E}\right) \) with \( E = \{ {xY} \mid x \in Y\} \) . Proof. Let \( P \) be any bipartite graph, with vertex classes \( \left\{ {{a}_{1},\ldots ,{a}_{n}}\right\} \) and \( \left\{ {{b}_{1},\ldots ,{b}_{m}}\right\} \), say. Let \( X \) be a set with \( {2n} + m \) elements, say \[ X = \left\{ {{x}_{1},\ldots ,{x}_{n},{y}_{1},\ldots ,{y}_{n},{z}_{1},\ldots ,{z}_{m}}\right\} \] we shall define an embedding \( \varphi : P \rightarrow \left( {X,{\left\lbrack X\right\rbrack }^{n + 1}, E}\right) \) . Let us start by setting \( \varphi \left( {a}_{i}\right) \mathrel{\text{:=}} {x}_{i} \) for all \( i = 1,\ldots, n \) . Which \( \left( {n + 1}\right) \) -sets \( Y \subseteq X \) are suitable candidates for the choice of \( \varphi \left( {b}_{i}\right) \) for a given vertex \( {b}_{i} \) ? Clearly those adjacent exactly to the images of the neighbours of \( {b}_{i} \), i.e. those satisfying \[ Y \cap \left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} = \varph

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

( {X,{\left\lbrack X\right\rbrack }^{k}, E}\right) \) with \( E = \{ {xY} \mid x \in Y\} \) . Proof. Let \( P \) be any bipartite graph, with vertex classes \( \left\{ {{a}_{1},\ldots ,{a}_{n}}\right\} \) and \( \left\{ {{b}_{1},\ldots ,{b}_{m}}\right\} \), say. Let \( X \) be a set with \( {2n} + m \) elements, say \[ X = \left\{ {{x}_{1},\ldots ,{x}_{n},{y}_{1},\ldots ,{y}_{n},{z}_{1},\ldots ,{z}_{m}}\right\} \] we shall define an embedding \( \varphi : P \rightarrow \left( {X,{\left\lbrack X\right\rbrack }^{n + 1}, E}\right) \) . Let us start by setting \( \varphi \left( {a}_{i}\right) \mathrel{\text{:=}} {x}_{i} \) for all \( i = 1,\ldots, n \) . Which \( \left( {n + 1}\right) \) -sets \( Y \subseteq X \) are suitable candidates for the choice of \( \varphi \left( {b}_{i}\right) \) for a given vertex \( {b}_{i} \) ? Clearly those adjacent exactly to the images of the neighbours of \( {b}_{i} \), i.e. those satisfying \[ Y \cap \left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} = \varphi \left( {{N}_{P}\left( {b}_{i}\right) }\right) . \] (1) Since \( d\left( {b}_{i}\right) \leq n \), the requirement of (1) leaves at least one of the \( n + 1 \) elements of \( Y \) unspecified. In addition to \( \varphi \left( {{N}_{P}\left( {b}_{i}\right) }\right) \), we may therefore include in each \( Y = \varphi \left( {b}_{i}\right) \) the vertex \( {z}_{i} \) as an ’index’; this ensures that \( \varphi \left( {b}_{i}\right) \neq \varphi \left( {b}_{j}\right) \) for \( i \neq j \), even when \( {b}_{i} \) and \( {b}_{j} \) have the same neighbours in \( P \) . To specify the sets \( Y = \varphi \left( {b}_{i}\right) \) completely, we finally fill them up with ’dummy’ elements \( {y}_{j} \) until \( \left| Y\right| = n + 1 \) . Our second lemma already covers the bipartite case of the theorem: it says that every bipartite graph has a Ramsey graph- even a bipartite one. Lemma 9.3.3. For every bipartite graph \( P \) there exists a bipartite graph \( {P}^{\prime } \) such that for every 2-colouring of the edges of \( {P}^{\prime } \) there is an embedding \( \varphi : P \rightarrow {P}^{\prime } \) for which all the edges of \( \varphi \left( P\right) \) have the same colour. --- (9.1.3) \( P, X, k, E \) \( {P}^{\prime },{X}^{\prime },{k}^{\prime } \) --- Proof. We may assume by Lemma 9.3.2 that \( P \) has the form \( \left( {X,{\left\lbrack X\right\rbrack }^{k}, E}\right) \) with \( E = \{ {xY} \mid x \in Y\} \) . We show the assertion for the graph \( {P}^{\prime } \mathrel{\text{:=}} \) \( \left( {{X}^{\prime },{\left\lbrack {X}^{\prime }\right\rbrack }^{{k}^{\prime }},{E}^{\prime }}\right) \), where \( {k}^{\prime } \mathrel{\text{:=}} {2k} - 1,{X}^{\prime } \) is any set of cardinality \[ \left| {X}^{\prime }\right| = R\left( {{k}^{\prime },2\left( \begin{matrix} {k}^{\prime } \\ k \end{matrix}\right), k\left| X\right| + k - 1}\right) , \] (this is the Ramsey number defined after Theorem 9.1.3), and \[ {E}^{\prime } \mathrel{\text{:=}} \left\{ {{x}^{\prime }{Y}^{\prime } \mid {x}^{\prime } \in {Y}^{\prime }}\right\} \] Let us then colour the edges of \( {P}^{\prime } \) with two colours \( \alpha \) and \( \beta \) . Of the \( \alpha ,\beta \) \( \left| {Y}^{\prime }\right| = {2k} - 1 \) edges incident with a vertex \( {Y}^{\prime } \in {\left\lbrack {X}^{\prime }\right\rbrack }^{{k}^{\prime }} \), at least \( k \) must have the same colour. For each \( {Y}^{\prime } \) we may therefore choose a fixed \( k \) -set \( {Z}^{\prime } \subseteq {Y}^{\prime } \) such that all the edges \( {x}^{\prime }{Y}^{\prime } \) with \( {x}^{\prime } \in {Z}^{\prime } \) have the same colour; \( {Z}^{\prime } \) we shall call this colour associated with \( {Y}^{\prime } \) . associated The sets \( {Z}^{\prime } \) can lie within their supersets \( {Y}^{\prime } \) in \( \left( \begin{matrix} {k}^{\prime } \\ k \end{matrix}\right) \) ways, as follows. Let \( {X}^{\prime } \) be linearly ordered. Then for every \( {Y}^{\prime } \in {\left\lbrack {X}^{\prime }\right\rbrack }^{{k}^{\prime }} \) there is a unique order-preserving bijection \( {\sigma }_{{Y}^{\prime }} : {Y}^{\prime } \rightarrow \left\{ {1,\ldots ,{k}^{\prime }}\right\} \), which maps \( {Z}^{\prime } \) to one \( {\sigma }_{{Y}^{\prime }} \) of \( \left( \begin{matrix} {k}^{\prime } \\ k \end{matrix}\right) \) possible images. We now colour \( {\left\lbrack {X}^{\prime }\right\rbrack }^{{k}^{\prime }} \) with the \( 2\left( \begin{matrix} {k}^{\prime } \\ k \end{matrix}\right) \) elements of the set \[ {\left\lbrack \left\{ 1,\ldots ,{k}^{\prime }\right\} \right\rbrack }^{k} \times \{ \alpha ,\beta \} \] as colours, giving each \( {Y}^{\prime } \in {\left\lbrack {X}^{\prime }\right\rbrack }^{{k}^{\prime }} \) as its colour the pair \( \left( {{\sigma }_{{Y}^{\prime }}\left( {Z}^{\prime }\right) ,\gamma }\right) \) , where \( \gamma \) is the colour \( \alpha \) or \( \beta \) associated with \( {Y}^{\prime } \) . Since \( \left| {X}^{\prime }\right| \) was chosen as the Ramsey number with parameters \( {k}^{\prime },2\left( \begin{matrix} {k}^{\prime } \\ k \end{matrix}\right) \) and \( k\left| X\right| + k - 1 \), we know that \( {X}^{\prime } \) has a monochromatic subset \( W \) of cardinality \( k\left| X\right| + k - 1 \) . \( W \) All \( {Z}^{\prime } \) with \( {Y}^{\prime } \subseteq W \) thus lie within their \( {Y}^{\prime } \) in the same way, i.e. there exists an \( S \in {\left\lbrack \left\{ 1,\ldots ,{k}^{\prime }\right\} \right\rbrack }^{k} \) such that \( {\sigma }_{{Y}^{\prime }}\left( {Z}^{\prime }\right) = S \) for all \( {Y}^{\prime } \in {\left\lbrack W\right\rbrack }^{{k}^{\prime }} \) , and all \( {Y}^{\prime } \in {\left\lbrack W\right\rbrack }^{{k}^{\prime }} \) are associated with the same colour, say with \( \alpha \) . \( \alpha \) We now construct the desired embedding \( \varphi \) of \( P \) in \( {P}^{\prime } \) . We first \( {\left. \varphi \right| }_{X} \) define \( \varphi \) on \( X = : \left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} \), choosing images \( \varphi \left( {x}_{i}\right) = : {w}_{i} \in W \) so \( {x}_{i},{w}_{i}, n \) that \( {w}_{i} < {w}_{j} \) in our ordering of \( {X}^{\prime } \) whenever \( i < j \) . Moreover, we choose the \( {w}_{i} \) so that exactly \( k - 1 \) elements of \( W \) are smaller than \( {w}_{1} \), exactly \( k - 1 \) lie between \( {w}_{i} \) and \( {w}_{i + 1} \) for \( i = 1,\ldots, n - 1 \), and exactly \( k - 1 \) are bigger than \( {w}_{n} \) . Since \( \left| W\right| = {kn} + k - 1 \), this can indeed be done (Fig. 9.3.4). We now define \( \varphi \) on \( {\left\lbrack X\right\rbrack }^{k} \) . Given \( Y \in {\left\lbrack X\right\rbrack }^{k} \), we wish to choose \( \varphi \left( Y\right) = : {Y}^{\prime } \in {\left\lbrack {X}^{\prime }\right\rbrack }^{{k}^{\prime }} \) so that the neighbours of \( {Y}^{\prime } \) among the vertices in \( \varphi \left( X\right) \) are precisely the images of the neighbours of \( Y \) in \( P \), i.e. the \( k \) vertices \( \varphi \left( x\right) \) with \( x \in Y \), and so that all these edges at \( {Y}^{\prime } \) are coloured \( \alpha \) . To find such a set \( {Y}^{\prime } \), we first fix its subset \( {Z}^{\prime } \) as \( \{ \varphi \left( x\right) \mid x \in Y\} \) (these are \( k \) vertices of type \( {w}_{i} \) ) and then extend \( {Z}^{\prime } \) by \( {k}^{\prime } - k \) further vertices \( u \in W \smallsetminus \varphi \left( X\right) \) to a set \( {Y}^{\prime } \in {\left\lbrack W\right\rbrack }^{{k}^{\prime }} \), in such a way that \( {Z}^{\prime } \) lies correctly within \( {Y}^{\prime } \), i.e. so that \( {\sigma }_{{Y}^{\prime }}\left( {Z}^{\prime }\right) = S \) . This can be done, because \( k - 1 = {k}^{\prime } - k \) other vertices of \( W \) lie between any two \( {w}_{i} \) . Then \[ {Y}^{\prime } \cap \varphi \left( X\right) = {Z}^{\prime } = \{ \varphi \left( x\right) \mid x \in Y\} \] so \( {Y}^{\prime } \) has the correct neighbours in \( \varphi \left( X\right) \), and all the edges between \( {Y}^{\prime } \) and these neighbours are coloured \( \alpha \) (because those neighbours lie in \( {Z}^{\prime } \) and \( {Y}^{\prime } \) is associated with \( \alpha \) ). Finally, \( \varphi \) is injective on \( {\left\lbrack X\right\rbrack }^{k} \) : the images \( {Y}^{\prime } \) of different vertices \( Y \) are distinct, because their intersections with \( \varphi \left( X\right) \) differ. Hence, our map \( \varphi \) is indeed an embedding of \( P \) in \( {P}^{\prime } \) .  Fig. 9.3.4. The graph of Lemma 9.3.3 Second proof of Theorem 9.3.1. Let \( H \) be given as in the theorem, and let \( n \mathrel{\text{:=}} R\left( r\right) \) be the Ramsey number of \( r \mathrel{\text{:=}} \left| H\right| \) . Then, for every \( K \) 2-colouring of its edges, the graph \( K = {K}^{n} \) contains a monochromatic copy of \( H \) -although not necessarily induced. We start by constructing a graph \( {G}^{0} \), as follows. Imagine the vertices of \( K \) to be arranged in a column, and replace every vertex by a row of \( \left( \begin{array}{l} n \\ r \end{array}\right) \) vertices. Then each of the \( \left( \begin{array}{l} n \\ r \end{array}\right) \) columns arising can be associated with one of the \( \left( \begin{array}{l} n \\ r \end{array}\right) \) ways of embedding \( V\left( H\right) \) in \( V\left( K\right) \) ; let us furnish this column with the edges of such a copy of \( H \) . The graph \( {G}^{0} \) thus arising consists of \( \left( \begin{array}{l} n \\ r \end{array}\right) \) disjoint copies of \( H \) and \( \left( {n - r}\right) \left( \begin{array}{l} n \\ r \end{array}\right) \) isolated vertices (Fig. 9.3.5). In order to define \( {G}^{0} \) formally, we assume that \( V\left( K\right) = \{ 1,\ldots, n\} \) and choose copies \( {H}_{1},\ldots ,{H}_{\left( n\right)

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ic copy of \( H \) -although not necessarily induced. We start by constructing a graph \( {G}^{0} \), as follows. Imagine the vertices of \( K \) to be arranged in a column, and replace every vertex by a row of \( \left( \begin{array}{l} n \\ r \end{array}\right) \) vertices. Then each of the \( \left( \begin{array}{l} n \\ r \end{array}\right) \) columns arising can be associated with one of the \( \left( \begin{array}{l} n \\ r \end{array}\right) \) ways of embedding \( V\left( H\right) \) in \( V\left( K\right) \) ; let us furnish this column with the edges of such a copy of \( H \) . The graph \( {G}^{0} \) thus arising consists of \( \left( \begin{array}{l} n \\ r \end{array}\right) \) disjoint copies of \( H \) and \( \left( {n - r}\right) \left( \begin{array}{l} n \\ r \end{array}\right) \) isolated vertices (Fig. 9.3.5). In order to define \( {G}^{0} \) formally, we assume that \( V\left( K\right) = \{ 1,\ldots, n\} \) and choose copies \( {H}_{1},\ldots ,{H}_{\left( n\right) } \) of \( H \) in \( K \) with pairwise distinct vertex sets. (Thus, on each \( r \) -set in \( V\left( K\right) \) we have one fixed copy \( {H}_{j} \) of \( H \) .) We then define \[ V\left( {G}^{0}\right) \mathrel{\text{:=}} \left\{ {\left( {i, j}\right) \mid i = 1,\ldots, n;j = 1,\ldots ,\left( \begin{array}{l} n \\ r \end{array}\right) }\right\} \] and  Fig. 9.3.5. The graph \( {G}^{0} \) \[ E\left( {G}^{0}\right) \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{j = 1}}^{\left( \begin{array}{l} n \\ r \end{array}\right) }\left\{ {\left( {i, j}\right) \left( {{i}^{\prime }, j}\right) \mid i{i}^{\prime } \in E\left( {H}_{j}\right) }\right\} . \] The idea of the proof now is as follows. Our aim is to reduce the general case of the theorem to the bipartite case dealt with in Lemma 9.3.3. Applying the lemma iteratively to all the pairs of rows of \( {G}^{0} \), we construct a very large graph \( G \) such that for every edge colouring of \( G \) there is an induced copy of \( {G}^{0} \) in \( G \) that is monochromatic on all the bipartite subgraphs induced by its pairs of rows, i.e. in which edges between the same two rows always have the same colour. The projection of this \( {G}^{0} \subseteq G \) to \( \{ 1,\ldots, n\} \) (by contracting its rows) then defines an edge colouring of \( K \) . (If the contraction does not yield all the edges of \( K \) , colour the missing edges arbitrarily.) By the choice of \( \left| K\right| \), some \( {K}^{r} \subseteq K \) will be monochromatic. The \( {H}_{j} \) inside this \( {K}^{r} \) then occurs with the same colouring in the \( j \) th column of our \( {G}^{0} \), where it is an induced subgraph of \( {G}^{0} \), and hence of \( G \) . Formally, we shall define a sequence \( {G}^{0},\ldots ,{G}^{m} \) of \( n \) -partite graphs \( {G}^{k} \), with \( n \) -partition \( \left\{ {{V}_{1}^{k},\ldots ,{V}_{n}^{k}}\right\} \) say, and then let \( G \mathrel{\text{:=}} {G}^{m} \) . The graph \( {G}^{0} \) has been defined above; let \( {V}_{1}^{0},\ldots ,{V}_{n}^{0} \) be its rows: \[ {V}_{i}^{0} \mathrel{\text{:=}} \left\{ {\left( {i, j}\right) \mid j = 1,\ldots ,\left( \begin{matrix} n \\ r \end{matrix}\right) }\right\} . \] Now let \( {e}_{1},\ldots ,{e}_{m} \) be an enumeration of the edges of \( K \) . For \( k = \) \( {e}_{k}, m \) \( 0,\ldots, m - 1 \), construct \( {G}^{k + 1} \) from \( {G}^{k} \) as follows. If \( {e}_{k + 1} = {i}_{1}{i}_{2} \), say, \( {i}_{1},{i}_{2} \) let \( P = \left( {{V}_{{i}_{1}}^{k},{V}_{{i}_{2}}^{k}, E}\right) \) be the bipartite subgraph of \( {G}^{k} \) induced by its \( {i}_{1} \) th and \( {i}_{2} \) th row. By Lemma 9.3.3, \( P \) has a bipartite Ramsey graph \( {P}^{\prime } \) \( {P}^{\prime } = \left( {{W}_{1},{W}_{2},{E}^{\prime }}\right) \) . We wish to define \( {G}^{k + 1} \supseteq {P}^{\prime } \) in such a way that every \( {W}_{1},{W}_{2} \) (monochromatic) embedding \( P \rightarrow {P}^{\prime } \) can be extended to an embedding \( {G}^{k} \rightarrow {G}^{k + 1} \) respecting their \( n \) -partitions. Let \( \left\{ {{\varphi }_{1},\ldots ,{\varphi }_{q}}\right\} \) be the set of \( {\varphi }_{p}, q \) all embeddings of \( P \) in \( {P}^{\prime } \), and let \[ V\left( {G}^{k + 1}\right) \mathrel{\text{:=}} {V}_{1}^{k + 1} \cup \ldots \cup {V}_{n}^{k + 1} \] where \[ {V}_{i}^{k + 1} \mathrel{\text{:=}} \left\{ \begin{array}{ll} {W}_{1} & \text{ for }i = {i}_{1} \\ {W}_{2} & \text{ for }i = {i}_{2} \\ \mathop{\bigcup }\limits_{{p = 1}}^{q}\left( {{V}_{i}^{k}\times \{ p\} }\right) & \text{ for }i \notin \left\{ {{i}_{1},{i}_{2}}\right\} . \end{array}\right. \] (Thus for \( i \neq {i}_{1},{i}_{2} \), we take as \( {V}_{i}^{k + 1} \) just \( q \) disjoint copies of \( {V}_{i}^{k} \) .) We now define the edge set of \( {G}^{k + 1} \) so that the obvious extensions of \( {\varphi }_{p} \) to all of \( V\left( {G}^{k}\right) \) become embeddings of \( {G}^{k} \) in \( {G}^{k + 1} \) : for \( p = 1,\ldots, q \), let \( {\psi }_{p} : V\left( {G}^{k}\right) \rightarrow V\left( {G}^{k + 1}\right) \) be defined by \[ {\psi }_{p}\left( v\right) \mathrel{\text{:=}} \left\{ \begin{array}{ll} {\varphi }_{p}\left( v\right) & \text{ for }v \in P \\ \left( {v, p}\right) & \text{ for }v \notin P \end{array}\right. \] and let \[ E\left( {G}^{k + 1}\right) \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{p = 1}}^{q}\left\{ {{\psi }_{p}\left( v\right) {\psi }_{p}\left( {v}^{\prime }\right) \mid v{v}^{\prime } \in E\left( {G}^{k}\right) }\right\} . \] Now for every 2-colouring of its edges, \( {G}^{k + 1} \) contains an induced copy \( {\psi }_{p}\left( {G}^{k}\right) \) of \( {G}^{k} \) whose edges in \( P \), i.e. those between its \( {i}_{1} \) th and \( {i}_{2} \) th row, have the same colour: just choose \( p \) so that \( {\varphi }_{p}\left( P\right) \) is the monochromatic induced copy of \( P \) in \( {P}^{\prime } \) that exists by Lemma 9.3.3. We claim that \( G \mathrel{\text{:=}} {G}^{m} \) satisfies the assertion of the theorem. So let a 2-colouring of the edges of \( G \) be given. By the construction of \( {G}^{m} \) from \( {G}^{m - 1} \), we can find in \( {G}^{m} \) an induced copy of \( {G}^{m - 1} \) such that for \( {e}_{m} = i{i}^{\prime } \) all edges between the \( i \) th and the \( {i}^{\prime } \) th row have the same colour. In the same way, we find inside this copy of \( {G}^{m - 1} \) an induced copy of \( {G}^{m - 2} \) whose edges between the \( i \) th and the \( {i}^{\prime } \) th row have the same colour also for \( i{i}^{\prime } = {e}_{m - 1} \) . Continuing in this way, we finally arrive at an induced copy of \( {G}^{0} \) in \( G \) such that, for each pair \( \left( {i,{i}^{\prime }}\right) \), all the edges between \( {V}_{i}^{0} \) and \( {V}_{{i}^{\prime }}^{0} \) have the same colour. As shown earlier, this \( {G}^{0} \) contains a monochromatic induced copy \( {H}_{j} \) of \( H \) . ## 9.4 Ramsey properties and connectivity According to Ramsey’s theorem, every large enough graph \( G \) has a very dense or a very sparse induced subgraph of given order, a \( {K}^{r} \) or \( \overline{{K}^{r}} \) . If we assume that \( G \) is connected, we can say a little more: Proposition 9.4.1. For every \( r \in \mathbb{N} \) there is an \( n \in \mathbb{N} \) such that every connected graph of order at least \( n \) contains \( {K}^{r},{K}_{1, r} \) or \( {P}^{r} \) as an induced subgraph. Proof. Let \( d + 1 \) be the Ramsey number of \( r \), let \( n \mathrel{\text{:=}} \frac{d}{d - 2}{\left( d - 1\right) }^{r} \) , (1.3.3) and let \( G \) be a graph of order at least \( n \) . If \( G \) has a vertex \( v \) of degree at least \( d + 1 \) then, by Theorem 9.1.1 and the choice of \( d \), either \( N\left( v\right) \) induces a \( {K}^{r} \) in \( G \) or \( \{ v\} \cup N\left( v\right) \) induces a \( {K}_{1, r} \) . On the other hand, if \( \Delta \left( G\right) \leq d \), then by Proposition 1.3.3 \( G \) has radius \( > r \), and hence contains two vertices at a distance \( \geq r \) . Any shortest path in \( G \) between these two vertices contains a \( {P}^{r} \) . In principle, we could now look for a similar set of 'unavoidable' \( k \) -connected subgraphs for any given connectivity \( k \) . To keep thse ’unavoidable sets' small, it helps to relax the containment relation from ’induced subgraph’ for \( k = 1 \) (as above) to ’topological minor’ for \( k = 2 \) , and on to ’minor’ for \( k = 3 \) and \( k = 4 \) . For larger \( k \), no similar results are known. Proposition 9.4.2. For every \( r \in \mathbb{N} \) there is an \( n \in \mathbb{N} \) such that every 2-connected graph of order at least \( n \) contains \( {C}^{r} \) or \( {K}_{2, r} \) as a topological minor. Proof. Let \( d \) be the \( n \) associated with \( r \) in Proposition 9.4.1, and let \( G \) be --- (3.3.6) --- a 2-connected graph with at least \( \frac{d}{d - 2}{\left( d - 1\right) }^{r} \) vertices. By Proposition 1.3.3, either \( G \) has a vertex of degree \( > d \) or \( \operatorname{diam}G \geq \operatorname{rad}G > r \) . In the latter case let \( a, b \in G \) be two vertices at distance \( > r \) . By Menger’s theorem (3.3.6), \( G \) contains two independent \( a - b \) paths. These form a cycle of length \( > r \) . Assume now that \( G \) has a vertex \( v \) of degree \( > d \) . Since \( G \) is 2- connected, \( G - v \) is connected and thus has a spanning tree; let \( T \) be a minimal tree in \( G - v \) that contains all the neighbours of \( v \) . Then every leaf of \( T \) is a neighbour of \( v \) . By the choice of \( d \), either \( T \) has a vertex of degree \( \geq r \) or \( T \) contains a path of length \( \geq r \), without loss of generality linking two leaves. Together with \( v \), such a path forms a cycle of length \( \geq r \) . A vertex \( u \) of degree \(

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onnected graph with at least \( \frac{d}{d - 2}{\left( d - 1\right) }^{r} \) vertices. By Proposition 1.3.3, either \( G \) has a vertex of degree \( > d \) or \( \operatorname{diam}G \geq \operatorname{rad}G > r \) . In the latter case let \( a, b \in G \) be two vertices at distance \( > r \) . By Menger’s theorem (3.3.6), \( G \) contains two independent \( a - b \) paths. These form a cycle of length \( > r \) . Assume now that \( G \) has a vertex \( v \) of degree \( > d \) . Since \( G \) is 2- connected, \( G - v \) is connected and thus has a spanning tree; let \( T \) be a minimal tree in \( G - v \) that contains all the neighbours of \( v \) . Then every leaf of \( T \) is a neighbour of \( v \) . By the choice of \( d \), either \( T \) has a vertex of degree \( \geq r \) or \( T \) contains a path of length \( \geq r \), without loss of generality linking two leaves. Together with \( v \), such a path forms a cycle of length \( \geq r \) . A vertex \( u \) of degree \( \geq r \) in \( T \) can be joined to \( v \) by \( r \) independent paths through \( T \), to form a \( T{K}_{2, r} \) . Theorem 9.4.3. (Oporowski, Oxley & Thomas 1993) For every \( r \in \mathbb{N} \) there is an \( n \in \mathbb{N} \) such that every 3-connected graph of order at least \( n \) contains a wheel of order \( r \) or a \( {K}_{3, r} \) as a minor. Let us call a graph of the form \( {C}^{n} * \overline{{K}^{2}}\left( {n \geq 4}\right) \) a double wheel, the 1-skeleton of a triangulation of the cylinder as in Fig. 9.4.1 a crown, and the 1-skeleton of a triangulation of the Möbius strip a Möbius crown.  Fig. 9.4.1. A crown and a Möbius crown Theorem 9.4.4. (Oporowski, Oxley & Thomas 1993) For every \( r \in \mathbb{N} \) there is an \( n \in \mathbb{N} \) such that every 4-connected graph with at least \( n \) vertices has a minor of order \( \geq r \) that is a double wheel, a crown, a Möbius crown, or a \( {K}_{4, s} \) . At first glance, the 'unavoidable' substructures presented in the four theorems above may seem to be chosen somewhat arbitrarily. In fact, the contrary is true: these sets are smallest possible, and as such unique. To make this precise, let us consider graph properties \( \mathcal{P} \) each containing arbitrarily large graphs. Given an order relation \( \leq \) between graphs (such as the subgraph relation \( \subseteq \), or the minor relation \( \preccurlyeq \) ), we write \( \mathcal{P} \leq {\mathcal{P}}^{\prime } \) if for every \( G \in \mathcal{P} \) there is a \( {G}^{\prime } \in {\mathcal{P}}^{\prime } \) such that \( G \leq {G}^{\prime } \) . If \( \mathcal{P} \leq {\mathcal{P}}^{\prime } \) as well as \( \mathcal{P} \geq {\mathcal{P}}^{\prime } \), we call \( \mathcal{P} \) and \( {\mathcal{P}}^{\prime } \) equivalent and write \( \mathcal{P} \sim {\mathcal{P}}^{\prime } \) . For example, if \( \leq \) is the subgraph relation, \( \mathcal{P} \) is the class of all paths, \( {\mathcal{P}}^{\prime } \) is the class of paths of even length, and \( \mathcal{S} \) is the class of all subdivisions of stars, then \( \mathcal{P} \sim {\mathcal{P}}^{\prime } \leq \mathcal{S} \nleq \mathcal{P} \) . --- Kuratowski set --- If \( \mathcal{C} \) is a collection of such properties, we call a finite subset \( \left\{ {{\mathcal{P}}_{1},\ldots ,{\mathcal{P}}_{k}}\right\} \) of \( \mathcal{C} \) a Kuratowski set for \( \mathcal{C} \) (with respect to \( \leq \) ) if the \( {\mathcal{P}}_{i} \) are incomparable (i.e., \( {\mathcal{P}}_{i} \nleq {\mathcal{P}}_{j} \) whenever \( i \neq j \) ) and for every \( \mathcal{P} \in \mathcal{C} \) unique there is an \( i \) such that \( {\mathcal{P}}_{i} \leq \mathcal{P} \) . We call this Kuratowski set unique if every Kuratowski set for \( \mathcal{C} \) can be written as \( \left\{ {{\mathcal{Q}}_{1},\ldots ,{\mathcal{Q}}_{k}}\right\} \) with \( {\mathcal{Q}}_{i} \sim {\mathcal{P}}_{i} \) for all \( i \) . The essence of our last four theorems can now be stated more comprehensively as follows (cf. Exercise 18). ## Theorem 9.4.5. (i) The stars and the paths form the unique (2-element) Kuratowski set for the properties of connected graphs, with respect to the subgraph relation. (ii) The cycles and the graphs \( {K}_{2, r}\left( {r \in \mathbb{N}}\right) \) form the unique (2- element) Kuratowski set for the properties of 2-connected graphs, with respect to the topological minor relation. (iii) The wheels and the graphs \( {K}_{3, r}\left( {r \in \mathbb{N}}\right) \) form the unique (2- element) Kuratowski set for the properties of 3-connected graphs, with respect to the minor relation. (iv) The double wheels, the crowns, the Möbius crowns, and the graphs \( {K}_{4, r}\left( {r \in \mathbb{N}}\right) \) form the unique (4-element) Kuratowski set for the properties of 4-connected graphs, with respect to the minor relation. ## Exercises 1. \( {}^{ - } \) Determine the Ramsey number \( R\left( 3\right) \) . 2. \( {}^{ - } \) Deduce the case \( k = 2 \) (but \( c \) arbitrary) of Theorem 9.1.3 directly from Theorem 9.1.1. 3. Can you improve the exponential upper bound on the Ramsey number \( R\left( n\right) \) for perfect graphs? 4. \( {}^{ + } \) Construct a graph on \( \mathbb{R} \) that has neither a complete nor an edgeless induced subgraph on \( \left| \mathbb{R}\right| = {2}^{{\aleph }_{0}} \) vertices. (So Ramsey’s theorem does not extend to uncountable sets.) 5.* Prove the edge version of the Erdős-Pósa theorem (2.3.2): there exists a function \( g : \mathbb{N} \rightarrow \mathbb{R} \) such that, given \( k \in \mathbb{N} \), every graph contains either \( k \) edge-disjoint cycles or a set of at most \( g\left( k\right) \) edges meeting all its cycles. (Hint. Consider in each component a normal spanning tree \( T \) . If \( T \) has many chords \( {xy} \), use any regular pattern of how the paths \( {xTy} \) intersect to find many edge-disjoint cycles.) 6. \( {}^{ + } \) Use Ramsey’s theorem to show that for any \( k,\ell \in \mathbb{N} \) there is an \( n \in \mathbb{N} \) such that every sequence of \( n \) distinct integers contains an increasing subsequence of length \( k + 1 \) or a decreasing subsequence of length \( \ell + 1 \) . Find an example showing that \( n > k\ell \) . Then prove the theorem of Erdős and Szekeres that \( n = k\ell + 1 \) will do. 7. Sketch a proof of the following theorem of Erdős and Szekeres: for every \( k \in \mathbb{N} \) there is an \( n \in \mathbb{N} \) such that among any \( n \) points in the plane, no three of them collinear, there are \( k \) points spanning a convex \( k \) -gon, i.e. such that none of them lies in the convex hull of the others. 8. Prove the following result of Schur: for every \( k \in \mathbb{N} \) there is an \( n \in \mathbb{N} \) such that, for every partition of \( \{ 1,\ldots, n\} \) into \( k \) sets, at least one of the subsets contains numbers \( x, y, z \) such that \( x + y = z \) . 9. Let \( \left( {X, \leq }\right) \) be a totally ordered set, and let \( G = \left( {V, E}\right) \) be the graph on \( V \mathrel{\text{:=}} {\left\lbrack X\right\rbrack }^{2} \) with \( E \mathrel{\text{:=}} \left\{ {\left( {x, y}\right) \left( {{x}^{\prime },{y}^{\prime }}\right) \mid x < y = {x}^{\prime } < {y}^{\prime }}\right\} \) . (i) Show that \( G \) contains no triangle. (ii) Show that \( \chi \left( G\right) \) will get arbitrarily large if \( \left| X\right| \) is chosen large enough. 10. A family of sets is called a \( \Delta \) -system if every two of the sets have the same intersection. Show that every infinite family of sets of the same finite cardinality contains an infinite \( \Delta \) -system. 11. Prove that for every \( r \in \mathbb{N} \) and every tree \( T \) there exists a \( k \in \mathbb{N} \) such that every graph \( G \) with \( \chi \left( G\right) \geq k \) and \( \omega \left( G\right) < r \) contains a subdivision of \( T \) in which no two branch vertices are adjacent in \( G \) (unless they are adjacent in \( T \) ). 12. Let \( m, n \in \mathbb{N} \), and assume that \( m - 1 \) divides \( n - 1 \) . Show that every tree \( T \) of order \( m \) satisfies \( R\left( {T,{K}_{1, n}}\right) = m + n - 1 \) . 13. Prove that \( {2}^{c} < R\left( {2, c,3}\right) \leq {3c} \) ! for every \( c \in \mathbb{N} \) . (Hint. Induction on \( c \) .) 14. \( {}^{ - } \) Derive the statement \( \left( *\right) \) in the first proof of Theorem 9.3.1 from the theorem itself, i.e. show that \( \left( *\right) \) is only formally stronger than the theorem. 15. Show that, given any two graphs \( {H}_{1} \) and \( {H}_{2} \), there exists a graph \( G = G\left( {{H}_{1},{H}_{2}}\right) \) such that, for every vertex-colouring of \( G \) with colours 1 and 2, there is either an induced copy of \( {H}_{1} \) coloured 1 or an induced copy of \( {\mathrm{H}}_{2} \) coloured 2 in \( G \) . 16. Show that the Ramsey graph \( G \) for \( H \) constructed in the second proof of Theorem 9.3.1 does indeed satisfy \( \omega \left( G\right) = \omega \left( H\right) \) . 17. \( {}^{ - } \) The \( {K}^{r} \) from Ramsey’s theorem, last sighted in Proposition 9.4.1, conspicuously fails to make an appearance from Proposition 9.4.2 onwards. Can it be excused? 18. Deduce Theorem 9.4.5 from the other four results in Section 9.4, and vice versa. ## Notes Due to increased interaction with research on random and pseudo-random \( {}^{4} \) structures (the latter being provided, for example, by the regularity lemma), the Ramsey theory of graphs has recently seen a period of major activity and advance. Theorem 9.2.2 is an early example of this development. For the more classical approach, the introductory text by R.L. Graham, B.L. Rothschild & J.H. Spencer, Ramsey Theory (2nd edn.), Wiley 1990, makes stimulating reading. This book includes a chapter on graph Ramsey t

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red 2 in \( G \) . 16. Show that the Ramsey graph \( G \) for \( H \) constructed in the second proof of Theorem 9.3.1 does indeed satisfy \( \omega \left( G\right) = \omega \left( H\right) \) . 17. \( {}^{ - } \) The \( {K}^{r} \) from Ramsey’s theorem, last sighted in Proposition 9.4.1, conspicuously fails to make an appearance from Proposition 9.4.2 onwards. Can it be excused? 18. Deduce Theorem 9.4.5 from the other four results in Section 9.4, and vice versa. ## Notes Due to increased interaction with research on random and pseudo-random \( {}^{4} \) structures (the latter being provided, for example, by the regularity lemma), the Ramsey theory of graphs has recently seen a period of major activity and advance. Theorem 9.2.2 is an early example of this development. For the more classical approach, the introductory text by R.L. Graham, B.L. Rothschild & J.H. Spencer, Ramsey Theory (2nd edn.), Wiley 1990, makes stimulating reading. This book includes a chapter on graph Ramsey theory, but is not confined to it. Surveys of finite and infinite Ramsey theory are given by J. Nešetřil and A. Hajnal in their chapters in the Handbook of Combinatorics (R.L. Graham, M. Grötschel & L. Lovász, eds.), North-Holland 1995. The Ramsey theory of infinite sets forms a substantial part of combinatorial set theory, and is treated in depth in P. Erdős, A. Hajnal, A. Máté & R. Rado, Combinatorial Set Theory, North-Holland 1984. An attractive collection of highlights from various branches of Ramsey theory, including applications in algebra, geometry and point-set topology, is offered in B. Bollobás, Graph Theory, Springer GTM 63, 1979. Theorem 9.2.2 is due to V. Chvátal, V. Rödl, E. Szemerédi & W.T. Trotter, The Ramsey number of a graph with bounded maximum degree, J. Com-bin. Theory B 34 (1983), 239-243. Our proof follows the sketch in J. Komlós & M. Simonovits, Szemerédi's Regularity Lemma and its applications in graph theory, in (D. Miklós, V.T. Sós & T. Szőnyi, eds.) Paul Erdős is 80, Vol. 2, Proc. Colloq. Math. Soc. János Bolyai (1996). The theorem marks a breakthrough towards a conjecture of Burr and Erdős (1975), which asserts that the --- \( {}^{4} \) Concrete graphs whose structure resembles the structure expected of a random graph are called pseudo-random. For example, the bipartite graphs spanned by an \( \epsilon \) -regular pair of vertex sets in a graph are pseudo-random. --- Ramsey numbers of graphs with bounded average degree in every subgraph are linear: for every \( d \in \mathbb{N} \), the conjecture says, there exists a constant \( c \) such that \( R\left( H\right) \leq c\left| H\right| \) for all graphs \( H \) with \( d\left( {H}^{\prime }\right) \leq d \) for all \( {H}^{\prime } \subseteq H \) . This conjecture has been verified approximately by A. Kostochka and B. Sudakov, On Ramsey numbers of sparse graphs, Combinatorics, Probability and Computing 12 (2003),627-641, who proved that \( R\left( H\right) \leq {\left| H\right| }^{1 + o\left( 1\right) } \) . Our first proof of Theorem 9.3.1 is based on W. Deuber, A generalization of Ramsey's theorem, in (A.Hajnal, R.Rado & V.T.Sós, eds.) Infnite and finite sets, North-Holland 1975. The same volume contains the alternative proof of this theorem by Erdős, Hajnal and Pósa. Rödl proved the same result in his MSc thesis at the Charles University, Prague, in 1973. Our second proof of Theorem 9.3.1, which preserves the clique number of \( H \) for \( G \), is due to J. Nešetřil & V. Rödl, A short proof of the existence of restricted Ramsey graphs by means of a partite construction, Combinatorica 1 (1981), 199-202. The two theorems in Section 9.4 are due to B. Oporowski, J. Oxley & R. Thomas, Typical subgraphs of 3- and 4-connected graphs, J. Combin. Theory B 57 (1993), 239-257. 10 ## Hamilton Cycles In Chapter 1.8 we briefly discussed the problem of when a graph contains an Euler tour, a closed walk traversing every edge exactly once. The simple Theorem 1.8.1 solved that problem quite satisfactorily. Let us now ask the analogous question for vertices: when does a graph \( G \) contain a closed walk that contains every vertex of \( G \) exactly once? If \( \left| G\right| \geq 3 \) , --- Hamilton cycle Hamilton path --- then any such walk is a cycle: a Hamilton cycle of \( G \) . If \( G \) has a Hamilton cycle, it is called hamiltonian. Similarly, a path in \( G \) containing every vertex of \( G \) is a Hamilton path. To determine whether or not a given graph has a Hamilton cycle is much harder than deciding whether it is Eulerian, and no good characterization is known \( {}^{1} \) of the graphs that do. We shall begin this chapter by presenting the standard sufficient conditions for the existence of a Hamilton cycle (Sections 10.1 and 10.2). The rest of the chapter is then devoted to the beautiful theorem of Fleischner that the 'square' of every 2-connected graph has a Hamilton cycle. This is one of the main results in the field of Hamilton cycles. The simple proof we present (due to Ríha) is still a little longer than other proofs in this book, but not difficult. ## 10.1 Simple sufficient conditions What kind of condition might be sufficient for the existence of a Hamilton cycle in a graph \( G \) ? Purely global assumptions, like high edge density, will not be enough: we cannot do without the local property that every vertex has at least two neighbours. But neither is any large (but constant) minimum degree sufficient: it is easy to find graphs without a Hamilton cycle whose minimum degree exceeds any given constant bound. The following classic result derives its significance from this background: --- 1 ...or indeed expected to exist; see the notes for details. --- Theorem 10.1.1. (Dirac 1952) Every graph with \( n \geq 3 \) vertices and minimum degree at least \( n/2 \) has a Hamilton cycle. Proof. Let \( G = \left( {V, E}\right) \) be a graph with \( \left| G\right| = n \geq 3 \) and \( \delta \left( G\right) \geq n/2 \) . Then \( G \) is connected: otherwise, the degree of any vertex in the smallest component \( C \) of \( G \) would be less than \( \left| C\right| \leq n/2 \) . Let \( P = {x}_{0}\ldots {x}_{k} \) be a longest path in \( G \) . By the maximality of \( P \) , all the neighbours of \( {x}_{0} \) and all the neighbours of \( {x}_{k} \) lie on \( P \) . Hence at least \( n/2 \) of the vertices \( {x}_{0},\ldots ,{x}_{k - 1} \) are adjacent to \( {x}_{k} \), and at least \( n/2 \) of these same \( k < n \) vertices \( {x}_{i} \) are such that \( {x}_{0}{x}_{i + 1} \in E \) . By the pigeon hole principle, there is a vertex \( {x}_{i} \) that has both properties, so we have \( {x}_{0}{x}_{i + 1} \in E \) and \( {x}_{i}{x}_{k} \in E \) for some \( i < k \) (Fig. 10.1.1).  Fig. 10.1.1. Finding a Hamilton cycle in the proof Theorem 10.1.1 We claim that the cycle \( C \mathrel{\text{:=}} {x}_{0}{x}_{i + 1}P{x}_{k}{x}_{i}P{x}_{0} \) is a Hamilton cycle of \( G \) . Indeed, since \( G \) is connected, \( C \) would otherwise have a neighbour in \( G - C \), which could be combined with a spanning path of \( C \) into a path longer than \( P \) . Theorem 10.1.1 is best possible in that we cannot replace the bound of \( n/2 \) with \( \lfloor n/2\rfloor \) : if \( n \) is odd and \( G \) is the union of two copies of \( {K}^{\lceil n/2\rceil } \) meeting in one vertex, then \( \delta \left( G\right) = \lfloor n/2\rfloor \) but \( \kappa \left( G\right) = 1 \), so \( G \) cannot have a Hamilton cycle. In other words, the high level of the bound of \( \delta \geq n/2 \) is needed to ensure, if nothing else, that \( G \) is 2-connected: a condition just as trivially necessary for hamiltonicity as a minimum degree of at least 2. It would seem, therefore, that prescribing some high (constant) value for \( \kappa \) rather than for \( \delta \) stands a better chance of implying hamiltonicity. However, this is not so: although every large enough \( k \) -connected graph contains a cycle of length at least \( {2k} \) (Ex. 16, Ch. 3), the graphs \( {K}_{k, n} \) show that this is already best possible. Slightly more generally, a graph \( G \) with a separating set \( S \) of \( k \) vertices such that \( G - S \) has more than \( k \) components is clearly not hamiltonian. Could it be true that all non-hamiltonian graphs have such a separating set, one that leaves many components compared with its size? We shall address this question in a moment. For now, just note that such graphs as above also have relatively large independent sets: pick one vertex from each component of \( G - S \) to obtain one of order at least \( k + 1 \) . Might we be able to force a Hamilton cycle by forbidding large independent sets? By itself, the assumption of \( \alpha \left( G\right) \leq k \) already guarantees a cycle of length at least \( \left| G\right| /k \) (Ex. 13, Ch. 5). And combined with the assumption of \( k \) -connectedness, it does indeed imply hamiltonicity: Proposition 10.1.2. Every graph \( G \) with \( \left| G\right| \geq 3 \) and \( \alpha \left( G\right) \leq \kappa \left( G\right) \) has a Hamilton cycle. Proof. Put \( \kappa \left( G\right) = : k \), and let \( C \) be a longest cycle in \( G \) . Enumerate the --- (3.3.4) --- vertices of \( C \) cyclically, say as \( V\left( C\right) = \left\{ {{v}_{i} \mid i \in {\mathbb{Z}}_{n}}\right\} \) with \( {v}_{i}{v}_{i + 1} \in E\left( C\right) \) for all \( i \in {\mathbb{Z}}_{n} \) . If \( C \) is not a Hamilton cycle, pick a vertex \( v \in G - C \) and a \( v - C \) fan \( \mathcal{F} = \left\{ {{P}_{i} \mid i \in I}\right\} \) in \( G \), where \( I \subseteq {\mathbb{Z}}_{n} \) and each \( {P}_{i} \) ends in \( {v}_{i} \) . Let \( \mathcal{F} \) be chosen with maximum cardinality; then \( v{v}_{j} \notin E\left( G\right) \) for any \( j \notin I \), and \[ \

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k \) (Ex. 13, Ch. 5). And combined with the assumption of \( k \) -connectedness, it does indeed imply hamiltonicity: Proposition 10.1.2. Every graph \( G \) with \( \left| G\right| \geq 3 \) and \( \alpha \left( G\right) \leq \kappa \left( G\right) \) has a Hamilton cycle. Proof. Put \( \kappa \left( G\right) = : k \), and let \( C \) be a longest cycle in \( G \) . Enumerate the --- (3.3.4) --- vertices of \( C \) cyclically, say as \( V\left( C\right) = \left\{ {{v}_{i} \mid i \in {\mathbb{Z}}_{n}}\right\} \) with \( {v}_{i}{v}_{i + 1} \in E\left( C\right) \) for all \( i \in {\mathbb{Z}}_{n} \) . If \( C \) is not a Hamilton cycle, pick a vertex \( v \in G - C \) and a \( v - C \) fan \( \mathcal{F} = \left\{ {{P}_{i} \mid i \in I}\right\} \) in \( G \), where \( I \subseteq {\mathbb{Z}}_{n} \) and each \( {P}_{i} \) ends in \( {v}_{i} \) . Let \( \mathcal{F} \) be chosen with maximum cardinality; then \( v{v}_{j} \notin E\left( G\right) \) for any \( j \notin I \), and \[ \left| \mathcal{F}\right| \geq \min \{ k,\left| C\right| \} \] (1) by Menger's theorem (3.3.4).  Fig. 10.1.2. Two cycles longer than \( C \) For every \( i \in I \), we have \( i + 1 \notin I \) : otherwise, \( \left( {C \cup {P}_{i} \cup {P}_{i + 1}}\right) - {v}_{i}{v}_{i + 1} \) would be a cycle longer than \( C \) (Fig. 10.1.2, left). Thus \( \left| \mathcal{F}\right| < \left| C\right| \), and hence \( \left| I\right| = \left| \mathcal{F}\right| \geq k \) by (1). Furthermore, \( {v}_{i + 1}{v}_{j + 1} \notin E\left( G\right) \) for all \( i, j \in I \) , as otherwise \( \left( {C \cup {P}_{i} \cup {P}_{j}}\right) + {v}_{i + 1}{v}_{j + 1} - {v}_{i}{v}_{i + 1} - {v}_{j}{v}_{j + 1} \) would be a cycle longer than \( C \) (Fig. 10.1.2, right). Hence \( \left\{ {{v}_{i + 1} \mid i \in I}\right\} \cup \{ v\} \) is a set of \( k + 1 \) or more independent vertices in \( G \), contradicting \( \alpha \left( G\right) \leq k \) . Let us return to the question whether an assumption that no small separator leaves many components can guarantee a Hamilton cycle. A graph \( G \) is called \( t \) -tough, where \( t > 0 \) is any real number, if for every \( t \) -tough separator \( S \) the graph \( G - S \) has at most \( \left| S\right| /t \) components. Clearly, hamiltonian graphs must be 1-tough - so what about the converse? Unfortunately, it is not difficult to find even small graphs that are 1-tough but have no Hamilton cycle (Exercise 5), so toughness does not provide a characterization of hamiltonian graphs in the spirit of Menger's theorem or Tutte's 1-factor theorem. However, a famous conjecture asserts that \( t \) -toughness for some \( t \) will force hamiltonicity: ## Toughness Conjecture. (Chvátal 1973) There exists an integer \( t \) such that every \( t \) -tough graph has a Hamilton cycle. The toughness conjecture was long expected to hold even with \( t = 2 \) . This has recently been disproved, but the general conjecture remains open. See the exercises for how the conjecture ties in with the results given in the remainder of this chapter. It may come as a surprise to learn that hamiltonicity is also related to the four colour problem. As we noted in Chapter 6.6, the four colour theorem is equivalent to the non-existence of a planar snark, i.e. to the assertion that every bridgeless planar cubic graph has a 4-flow. It is easily checked that 'bridgeless' can be replaced with '3-connected' in this assertion, and that every hamiltonian graph has a 4-flow (Ex. 12, Ch. 6). For a proof of the four colour theorem, therefore, it would suffice to show that every 3-connected planar cubic graph has a Hamilton cycle! Unfortunately, this is not the case: the first counterexample was found by Tutte in 1946. Ten years later, Tutte proved the following deep theorem as a best possible weakening: Theorem 10.1.3. (Tutte 1956) Every 4-connected planar graph has a Hamilton cycle. Although, at first glance, it appears that the study of Hamilton cycles is a part of graph theory that cannot possibly extend to infinite graphs, there is a fascinating conjecture that does just that. Recall that a circle in an infinite graph \( G \) is a homeomorphic copy of the unit circle \( {S}^{1} \) --- Hamilton circle --- in the topological space \( \left| G\right| \) formed by \( G \) and its ends (see Chapter 8.5). A Hamilton circle of \( G \) is a circle that contains every vertex of \( G \) . Conjecture. (Bruhn 2003) Every locally finite 4-connected planar graph has a Hamilton circle. ## 10.2 Hamilton cycles and degree sequences Historically, Dirac's theorem formed the point of departure for the discovery of a series of weaker and weaker degree conditions, all sufficient for hamiltonicity. The development culminated in a single theorem that encompasses all the earlier results: the theorem we shall prove in this section. --- degree sequence --- If \( G \) is a graph with \( n \) vertices and degrees \( {d}_{1} \leq \ldots \leq {d}_{n} \), then the \( n \) -tuple \( \left( {{d}_{1},\ldots ,{d}_{n}}\right) \) is called the degree sequence of \( G \) . Note that this sequence is unique, even though \( G \) has several vertex enumerations giving rise to its degree sequence. Let us call an arbitrary integer sequence \( \left( {{a}_{1},\ldots ,{a}_{n}}\right) \) hamiltonian if every graph with \( n \) vertices and a degree hamiltonian sequence pointwise greater than \( \left( {{a}_{1},\ldots ,{a}_{n}}\right) \) is hamiltonian. (A sequence --- sequence pointwise greater --- \( \left( {{d}_{1},\ldots ,{d}_{n}}\right) \) is pointwise greater than \( \left( {{a}_{1},\ldots ,{a}_{n}}\right) \) if \( {d}_{i} \geq {a}_{i} \) for all \( i \) .) The following theorem characterizes all hamiltonian sequences: ## Theorem 10.2.1. (Chvátal 1972) An integer sequence \( \left( {{a}_{1},\ldots ,{a}_{n}}\right) \) such that \( 0 \leq {a}_{1} \leq \ldots \leq {a}_{n} < n \) and \( n \geq 3 \) is hamiltonian if and only if the following holds for every \( i < n/2 \) : \[ {a}_{i} \leq i \Rightarrow {a}_{n - i} \geq n - i. \] Proof. Let \( \left( {{a}_{1},\ldots ,{a}_{n}}\right) \) be an arbitrary integer sequence such that \( \left( {{a}_{1},\ldots ,{a}_{n}}\right) \) \( 0 \leq {a}_{1} \leq \ldots \leq {a}_{n} < n \) and \( n \geq 3 \) . We first assume that this sequence satisfies the condition of the theorem and prove that it is hamiltonian. Suppose not. Then there exists a graph whose degree sequence \( \left( {{d}_{1},\ldots ,{d}_{n}}\right) \) satisfies \( \left( {{d}_{1},\ldots ,{d}_{n}}\right) \) \[ {d}_{i} \geq {a}_{i}\;\text{ for all }i \] (1) but which has no Hamilton cycle. Let \( G = \left( {V, E}\right) \) be such a graph, \( G = \left( {V, E}\right) \) chosen with the maximum number of edges. By (1), our assumptions for \( \left( {{a}_{1},\ldots ,{a}_{n}}\right) \) transfer to the degree sequence \( \left( {{d}_{1},\ldots ,{d}_{n}}\right) \) of \( G \) ; thus, \[ {d}_{i} \leq i \Rightarrow {d}_{n - i} \geq n - i\;\text{ for all }i < n/2. \] (2) Let \( x, y \) be distinct and non-adjacent vertices in \( G \), with \( d\left( x\right) \leq d\left( y\right) \) \( x, y \) and \( d\left( x\right) + d\left( y\right) \) as large as possible. One easily checks that the degree sequence of \( G + {xy} \) is pointwise greater than \( \left( {{d}_{1},\ldots ,{d}_{n}}\right) \), and hence than \( \left( {{a}_{1},\ldots ,{a}_{n}}\right) \) . Hence, by the maximality of \( G \), the new edge \( {xy} \) lies on a Hamilton cycle \( H \) of \( G + {xy} \) . Then \( H - {xy} \) is a Hamilton path \( {x}_{1},\ldots ,{x}_{n} \) \( {x}_{1},\ldots ,{x}_{n} \) in \( G \), with \( {x}_{1} = x \) and \( {x}_{n} = y \) say. As in the proof of Dirac's theorem, we now consider the index sets \[ I \mathrel{\text{:=}} \left\{ {i \mid x{x}_{i + 1} \in E}\right\} \text{ and }J \mathrel{\text{:=}} \left\{ {j \mid {x}_{j}y \in E}\right\} . \] Then \( I \cup J \subseteq \{ 1,\ldots, n - 1\} \), and \( I \cap J = \varnothing \) because \( G \) has no Hamilton cycle. Hence \[ d\left( x\right) + d\left( y\right) = \left| I\right| + \left| J\right| < n, \] (3) so \( h \mathrel{\text{:=}} d\left( x\right) < n/2 \) by the choice of \( x \) . Since \( {x}_{i}y \notin E \) for all \( i \in I \), all these \( {x}_{i} \) were candidates for the choice of \( x \) (together with \( y \) ). Our choice of \( \{ x, y\} \) with \( d\left( x\right) + d\left( y\right) \) maximum thus implies that \( d\left( {x}_{i}\right) \leq d\left( x\right) \) for all \( i \in I \) . Hence \( G \) has at least \( \left| I\right| = h \) vertices of degree at most \( h \), so \( {d}_{h} \leq h \) . By (2), this implies that \( {d}_{n - h} \geq n - h \), i.e. the \( h + 1 \) vertices with the degrees \( {d}_{n - h},\ldots ,{d}_{n} \) all have degree at least \( n - h \) . Since \( d\left( x\right) = h \), one of these vertices, \( z \) say, is not adjacent to \( x \) . Since \[ d\left( x\right) + d\left( z\right) \geq h + \left( {n - h}\right) = n, \] this contradicts the choice of \( x \) and \( y \) by (3). Let us now show that, conversely, for every sequence \( \left( {{a}_{1},\ldots ,{a}_{n}}\right) \) as in the theorem, but with \[ {a}_{h} \leq h\;\text{ and }\;{a}_{n - h} \leq n - h - 1 \] for some \( h < n/2 \), there exists a graph that has a pointwise greater degree sequence than \( \left( {{a}_{1},\ldots ,{a}_{n}}\right) \) but no Hamilton cycle. As the sequence \[ \left( {\underset{h\text{ times }}{\underbrace{h,\ldots, h}},\underset{n - {2h}\text{ times }}{\underbrace{n - h - 1,\ldots, n - h - 1}},\underset{h\text{ times }}{\underbrace{n - 1,\ldots, n - 1}}}\right) \] is pointwise greater than \( \left( {{a}_{1},\ldots ,{a}_{n}}\right) \), it suffices to find a graph with this degree sequence that has no Hamilton cycle. ![ecdfe8f9-7805-48fc-b1e8-df5c40de94

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t) = h \), one of these vertices, \( z \) say, is not adjacent to \( x \) . Since \[ d\left( x\right) + d\left( z\right) \geq h + \left( {n - h}\right) = n, \] this contradicts the choice of \( x \) and \( y \) by (3). Let us now show that, conversely, for every sequence \( \left( {{a}_{1},\ldots ,{a}_{n}}\right) \) as in the theorem, but with \[ {a}_{h} \leq h\;\text{ and }\;{a}_{n - h} \leq n - h - 1 \] for some \( h < n/2 \), there exists a graph that has a pointwise greater degree sequence than \( \left( {{a}_{1},\ldots ,{a}_{n}}\right) \) but no Hamilton cycle. As the sequence \[ \left( {\underset{h\text{ times }}{\underbrace{h,\ldots, h}},\underset{n - {2h}\text{ times }}{\underbrace{n - h - 1,\ldots, n - h - 1}},\underset{h\text{ times }}{\underbrace{n - 1,\ldots, n - 1}}}\right) \] is pointwise greater than \( \left( {{a}_{1},\ldots ,{a}_{n}}\right) \), it suffices to find a graph with this degree sequence that has no Hamilton cycle.  Fig. 10.2.1. Any cycle containing \( {v}_{1},\ldots ,{v}_{h} \) misses \( {v}_{h + 1} \) Figure 10.2.1 shows such a graph, with vertices \( {v}_{1},\ldots ,{v}_{n} \) and the edge set \[ \left\{ {{v}_{i}{v}_{j} \mid i, j > h}\right\} \cup \left\{ {{v}_{i}{v}_{j} \mid i \leq h;j > n - h}\right\} ; \] it is the union of a \( {K}^{n - h} \) on the vertices \( {v}_{h + 1},\ldots ,{v}_{n} \) and a \( {K}_{h, h} \) with partition sets \( \left\{ {{v}_{1},\ldots ,{v}_{h}}\right\} \) and \( \left\{ {{v}_{n - h + 1},\ldots ,{v}_{n}}\right\} \) . By applying Theorem 10.2.1 to \( G * {K}^{1} \), one can easily prove the following adaptation of the theorem to Hamilton paths. Let an integer sequence be called path-hamiltonian if every graph with a pointwise greater degree sequence has a Hamilton path. Corollary 10.2.2. An integer sequence \( \left( {{a}_{1},\ldots ,{a}_{n}}\right) \) such that \( n \geq 2 \) and \( 0 \leq {a}_{1} \leq \ldots \leq {a}_{n} < n \) is path-hamiltonian if and only if every \( i \leq n/2 \) is such that \( {a}_{i} < i \Rightarrow {a}_{n + 1 - i} \geq n - i \) . ## 10.3 Hamilton cycles in the square of a graph Given a graph \( G \) and a positive integer \( d \), we denote by \( {G}^{d} \) the graph on \( V\left( G\right) \) in which two vertices are adjacent if and only if they have distance at most \( d \) in \( G \) . Clearly, \( G = {G}^{1} \subseteq {G}^{2} \subseteq \ldots \) Our goal in this section is to prove the following fundamental result: Theorem 10.3.1. (Fleischner 1974) If \( G \) is a 2-connected graph, then \( {G}^{2} \) has a Hamilton cycle. We begin with three simple lemmas. Let us say that an edge \( e \in {G}^{2} \) bridges a vertex \( v \in G \) if its ends are neighbours of \( v \) in \( G \) . bridges Lemma 10.3.2. Let \( P = {v}_{0}\ldots {v}_{k} \) be a path \( \left( {k \geq 1}\right) \), and let \( G \) be the graph obtained from \( P \) by adding two vertices \( u, w \), together with the edges \( u{v}_{1} \) and \( w{v}_{k} \) (Fig. 10.3.1). (i) \( {P}^{2} \) contains a path \( Q \) from \( {v}_{0} \) to \( {v}_{1} \) with \( V\left( Q\right) = V\left( P\right) \) and \( {v}_{k - 1}{v}_{k} \in E\left( Q\right) \), such that each of the vertices \( {v}_{1},\ldots ,{v}_{k - 1} \) is bridged by an edge of \( Q \) . (ii) \( {G}^{2} \) contains disjoint paths \( Q \) from \( {v}_{0} \) to \( {v}_{k} \) and \( {Q}^{\prime } \) from \( u \) to \( w \) , such that \( V\left( Q\right) \cup V\left( {Q}^{\prime }\right) = V\left( G\right) \) and each of the vertices \( {v}_{1},\ldots ,{v}_{k} \) is bridged by an edge of \( Q \) or \( {Q}^{\prime } \) .  Fig. 10.3.1. The graph \( G \) in Lemma 10.3.2 Proof. (i) If \( k \) is even, let \( Q \mathrel{\text{:=}} {v}_{0}{v}_{2}\ldots {v}_{k - 2}{v}_{k}{v}_{k - 1}{v}_{k - 3}\ldots {v}_{3}{v}_{1} \) . If \( k \) is odd, let \( Q \mathrel{\text{:=}} {v}_{0}{v}_{2}\ldots {v}_{k - 1}{v}_{k}{v}_{k - 2}\ldots {v}_{3}{v}_{1} \) . (ii) If \( k \) is even, let \( Q \mathrel{\text{:=}} {v}_{0}{v}_{2}\ldots {v}_{k - 2}{v}_{k} \) ; if \( k \) is odd, let \( Q \mathrel{\text{:=}} \) \( {v}_{0}{v}_{1}{v}_{3}\ldots {v}_{k - 2}{v}_{k} \) . In both cases, let \( {Q}^{\prime } \) be the \( u - w \) path on the remaining vertices of \( {G}^{2} \) . Lemma 10.3.3. Let \( G = \left( {V, E}\right) \) be a cubic multigraph with a Hamilton cycle \( C \) . Let \( e \in E\left( C\right) \) and \( f \in E \smallsetminus E\left( C\right) \) be edges with a common end \( v \) (Fig. 10.3.2). Then there exists a closed walk in \( G \) that traverses \( e \) once, every other edge of \( C \) once or twice, and every edge in \( E \smallsetminus E\left( C\right) \) once. This walk can be chosen to contain the triple \( \left( {e, v, f}\right) \), that is, it traverses \( e \) in the direction of \( v \) and then leaves \( v \) by the edge \( f \) .  Fig. 10.3.2. The multigraphs \( G \) and \( {G}^{\prime } \) in Lemma 10.3.3 Proof. By Proposition 1.2.1, \( C \) has even length. Replace every other edge of \( C \) by a double edge, in such a way that \( e \) does not get replaced. In the arising 4-regular multigraph \( {G}^{\prime } \), split \( v \) into two vertices \( {v}^{\prime },{v}^{\prime \prime } \) , making \( {v}^{\prime } \) incident with \( e \) and \( f \), and \( {v}^{\prime \prime } \) incident with the other two edges at \( v \) (Fig. 10.3.2). By Theorem 1.8.1 this multigraph has an Euler tour, which induces the desired walk in \( G \) . Lemma 10.3.4. For every 2-connected graph \( G \) and \( x \in V\left( G\right) \), there is a cycle \( C \subseteq G \) that contains \( x \) as well as a vertex \( y \neq x \) with \( {N}_{G}\left( y\right) \subseteq V\left( C\right) \) . Proof. If \( G \) has a Hamilton cycle, there is nothing more to show. If not, let \( {C}^{\prime } \subseteq G \) be any cycle containing \( x \) ; such a cycle exists, since \( G \) is 2-connected. Let \( D \) be a component of \( G - {C}^{\prime } \) . Assume that \( {C}^{\prime } \) and \( D \) are chosen so that \( \left| D\right| \) is minimal. Since \( G \) is 2-connected, \( D \) has at least two neighbours on \( {C}^{\prime } \) . Then \( {C}^{\prime } \) contains a path \( P \) between two such neighbours \( u \) and \( v \), whose interior \( \overset{ \circ }{P} \) does not contain \( x \) and has no neighbour in \( D \) (Fig. 10.3.3). Replacing \( P \) in \( {C}^{\prime } \) by a \( u - v \) path  Fig. 10.3.3. The proof of Lemma 10.3.4 through \( D \), we obtain a cycle \( C \) that contains \( x \) and a vertex \( y \in D \) . If \( y \) had a neighbour \( z \) in \( G - C \), then \( z \) would lie in a component \( {D}^{\prime } \subsetneqq D \) of \( G - C \), contradicting the choice of \( {C}^{\prime } \) and \( D \) . Hence all the neighbours of \( y \) lie on \( C \), and \( C \) satisfies the assertion of the lemma. Proof of Theorem 10.3.1. We show by induction on \( \left| G\right| \) that, given any vertex \( {x}^{ * } \in G \), there is a Hamilton cycle \( H \) in \( {G}^{2} \) with the following property: \[ \text{Both edges of}H\text{at}{x}^{ * }\text{lie in}G\text{.} \] \( \left( *\right) \) For \( \left| G\right| = 3 \) we have \( G = {K}^{3} \), and the assertion is trivial. So let \( \left| G\right| \geq 4 \), assume the assertion for graphs of smaller order, and let \( {x}^{ * } \in V\left( G\right) \) be given. By Lemma 10.3.4, there is a cycle \( C \subseteq G \) that \( {x}^{ * } \) contains both \( {x}^{ * } \) and a vertex \( {y}^{ * } \neq {x}^{ * } \) whose neighbours in \( G \) all lie \( {y}^{ * } \) on \( C \) . \( C \) If \( C \) is a Hamilton cycle of \( G \), there is nothing to show; so assume that \( G - C \neq \varnothing \) . Consider a component \( D \) of \( G - C \) . Let \( \widetilde{D} \) denote the graph \( G/\left( {G - D}\right) \) obtained from \( G \) by contracting \( G - D \) into a new vertex \( \widetilde{x} \) . If \( \left| D\right| = 1 \), set \( \mathcal{P}\left( D\right) \mathrel{\text{:=}} \{ D\} \) . If \( \left| D\right| > 1 \), then \( \widetilde{D} \) is again \( \mathcal{P}\left( D\right) \) 2-connected. Hence, by the induction hypothesis, \( {\widetilde{D}}^{2} \) has a Hamilton cycle \( \widetilde{C} \) whose edges at \( \widetilde{x} \) both lie in \( \widetilde{D} \) . Note that the path \( \widetilde{C} - \widetilde{x} \) may have some edges that do not lie in \( {G}^{2} \) : edges joining two neighbours of \( \widetilde{x} \) that have no common neighbour in \( G \) (and are themselves non-adjacent in \( G \) ). Let \( \widetilde{E} \) denote the set of these edges, and let \( \mathcal{P}\left( D\right) \) denote the set \( \mathcal{P}\left( D\right) \) of components of \( \left( {\widetilde{C} - \widetilde{x}}\right) - \widetilde{E} \) ; this is a set of paths in \( {G}^{2} \) whose ends are adjacent to \( \widetilde{x} \) in \( \widetilde{D} \) (Fig. 10.3.4).  Fig. 10.3.4. \( \mathcal{P}\left( D\right) \) consists of three paths, one of which is trivial Let \( \mathcal{P} \) denote the union of the sets \( \mathcal{P}\left( D\right) \) over all components \( D \) of \( G - C \) . Clearly, \( \mathcal{P} \) has the following properties: The elements of \( \mathcal{P} \) are pairwise disjoint paths in \( {G}^{2} \) avoid- ing \( C \), and \( V\left( G\right) = V\left( C\right) \cup \mathop{\bigcup }\limits_{{P \in \mathcal{P}}}V\left( P\right) \) . Every end \( y \) of a (1) path \( P \in \mathcal{P} \)

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

\) denote the set of these edges, and let \( \mathcal{P}\left( D\right) \) denote the set \( \mathcal{P}\left( D\right) \) of components of \( \left( {\widetilde{C} - \widetilde{x}}\right) - \widetilde{E} \) ; this is a set of paths in \( {G}^{2} \) whose ends are adjacent to \( \widetilde{x} \) in \( \widetilde{D} \) (Fig. 10.3.4).  Fig. 10.3.4. \( \mathcal{P}\left( D\right) \) consists of three paths, one of which is trivial Let \( \mathcal{P} \) denote the union of the sets \( \mathcal{P}\left( D\right) \) over all components \( D \) of \( G - C \) . Clearly, \( \mathcal{P} \) has the following properties: The elements of \( \mathcal{P} \) are pairwise disjoint paths in \( {G}^{2} \) avoid- ing \( C \), and \( V\left( G\right) = V\left( C\right) \cup \mathop{\bigcup }\limits_{{P \in \mathcal{P}}}V\left( P\right) \) . Every end \( y \) of a (1) path \( P \in \mathcal{P} \) has a neighbour on \( C \) in \( G \) ; we choose such a neighbour and call it the foot of \( P \) at \( y \) . If \( P \in \mathcal{P} \) is trivial, then \( P \) has exactly one foot. If \( P \) is non-trivial, then \( P \) has a foot at each of its ends. These two feet need not be distinct, however; so any non-trivial \( P \) has either one or two feet. We shall now modify \( \mathcal{P} \) a little, preserving the properties summarized under (1); no properties of \( \mathcal{P} \) other than those will be used later in the proof. If a vertex of \( C \) is a foot of two distinct paths \( P,{P}^{\prime } \in \mathcal{P} \), say at \( y \in P \) and at \( {y}^{\prime } \in {P}^{\prime } \), then \( y{y}^{\prime } \) is an edge and \( {Py}{y}^{\prime }{P}^{\prime } \) is a path in \( {G}^{2} \) ; we replace \( P \) and \( {P}^{\prime } \) in \( \mathcal{P} \) by this path. We repeat this modification of \( \mathcal{P} \) until the following holds: No vertex of \( C \) is a foot of two distinct paths in \( \mathcal{P} \) . (2) \( {\mathcal{P}}_{1},{\mathcal{P}}_{2} \) \( {\mathcal{P}}_{2}\; \) For \( \;i = 1,2 \) let \( {\mathcal{P}}_{i} \subseteq \mathcal{P} \) denote the set of all paths in \( \mathcal{P} \) with exactly \( i \) \( {X}_{1},{X}_{2} \) feet, and let \( {X}_{i} \subseteq V\left( C\right) \) denote the set of all feet of paths in \( {\mathcal{P}}_{i} \) . Then \( {X}_{1} \cap {X}_{2} = \varnothing \) by (2), and \( {y}^{ * } \notin {X}_{1} \cup {X}_{2} \) . Let us also simplify \( G \) a little; again, these changes will affect neither the paths in \( \mathcal{P} \) nor the validity of (1) and (2). First, we shall assume from now on that all elements of \( \mathcal{P} \) are paths in \( G \) itself, not just in \( {G}^{2} \) . This assumption may give us some additional edges for \( {G}^{2} \), but we shall not use these in our construction of the desired Hamilton cycle \( H \) . (Indeed, \( H \) will contain all the paths from \( \mathcal{P} \) whole, as subpaths.) Thus if \( H \) lies in \( {G}^{2} \) and satisfies \( \left( *\right) \) for the modified version of \( G \), it will do so also for the original. For every \( P \in \mathcal{P} \), we further delete all \( P - C \) edges in \( G \) except those between the ends of \( P \) and its corresponding feet. Finally, we delete all chords of \( C \) in \( G \) . We are thus assuming without loss of generality: The only edges of \( G \) between \( C \) and a path \( P \in \mathcal{P} \) are the two edges between the ends of \( P \) and its corresponding (3) feet. (If \( \left| P\right| = 1 \), these two edges coincide.) The only edges of \( G \) with both ends on \( C \) are the edges of \( C \) itself. Our goal is to construct the desired Hamilton cycle \( H \) of \( {G}^{2} \) from the paths in \( \mathcal{P} \) and suitable paths in \( {C}^{2} \) . As a first approximation, we shall construct a closed walk \( W \) in the graph \[ \widetilde{G} \mathrel{\text{:=}} G - \bigcup {\mathcal{P}}_{1} \] a walk that will already satisfy a \( \left( *\right) \) -type condition and traverse every path in \( {\mathcal{P}}_{2} \) exactly once. Later, we shall modify \( W \) so that it passes through every vertex of \( C \) exactly once and, finally, so as to include the paths from \( {\mathcal{P}}_{1} \) . For the construction of \( W \) we assume that \( {\mathcal{P}}_{2} \neq \varnothing \) ; the case of \( {\mathcal{P}}_{2} = \varnothing \) is much simpler and will be treated later. We start by choosing a fixed cyclic orientation of \( C \), a bijection \( i \mapsto {v}_{i} \) from \( {\mathbb{Z}}_{\left| C\right| } \) to \( V\left( C\right) \) with \( {v}_{i}{v}_{i + 1} \in E\left( C\right) \) for all \( i \in {\mathbb{Z}}_{\left| C\right| } \) . Let us think of this orientation as clockwise; then every vertex \( {v}_{i} \in C \) has a right neighbour \( {v}_{i}^{ + } \mathrel{\text{:=}} {v}_{i + 1} \) and a left neighbour \( {v}_{i}^{ - } \mathrel{\text{:=}} {v}_{i - 1} \) . Accordingly, the \( {v}^{ + } \), right edge \( {v}^{ - }v \) lies to the left of \( v \), the edge \( v{v}^{ + } \) lies on its right, and so on. \( {v}^{ - } \), left A non-trivial path \( P = {v}_{i}{v}_{i + 1}\ldots {v}_{j - 1}{v}_{j} \) in \( C \) such that \( V\left( P\right) \cap {X}_{2} = \) \( \left\{ {{v}_{i},{v}_{j}}\right\} \) will be called an interval, with left end \( {v}_{i} \) and right end \( {v}_{j} \) . interval Thus, \( C \) is the union of \( \left| {X}_{2}\right| = 2\left| {\mathcal{P}}_{2}\right| \) intervals. As usual, we write \( P = \) : \( \left\lbrack {{v}_{i},{v}_{j}}\right\rbrack \) and set \( \left( {{v}_{i},{v}_{j}}\right) \mathrel{\text{:=}} P \) as well as \( \left\lbrack {{v}_{i},{v}_{j}}\right) \mathrel{\text{:=}} P{v}_{j} \) and \( \left( {{v}_{i},{v}_{j}}\right\rbrack \mathrel{\text{:=}} {\mathring{v}}_{i}P \) . \( \left\lbrack {v, w}\right\rbrack \) etc. For intervals \( \left\lbrack {u, v}\right\rbrack \) and \( \left\lbrack {v, w}\right\rbrack \) with a common end \( v \) we say that \( \left\lbrack {u, v}\right\rbrack \) lies to the left of \( \left\lbrack {v, w}\right\rbrack \), and \( \left\lbrack {v, w}\right\rbrack \) lies to the right of \( \left\lbrack {u, v}\right\rbrack \) . We denote the unique interval \( \left\lbrack {v, w}\right\rbrack \) with \( {x}^{ * } \in (v, w\rbrack \) as \( {I}^{ * } \), the path in \( {\mathcal{P}}_{2} \) with \( {I}^{ * },{P}^{ * } \) foot \( w \) as \( {P}^{ * } \), and the path \( {I}^{ * }w{P}^{ * } \) as \( {Q}^{ * } \) . \( {Q}^{ * } \) For the construction of \( W \), we may think of \( \widetilde{G} \) as a multigraph \( M \) on \( {X}_{2} \) whose edges are the intervals on \( C \) and the paths in \( {\mathcal{P}}_{2} \) (with their feet as ends). By (2), \( M \) is cubic, so we may apply Lemma 10.3.3 with \( e \mathrel{\text{:=}} {I}^{ * } \) and \( f \mathrel{\text{:=}} {P}^{ * } \) . The lemma provides us with a closed walk \( W \) in \( \widetilde{G} \) which traverses \( {I}^{ * } \) once, every other interval of \( C \) once or twice, and every path in \( {\mathcal{P}}_{2} \) once. Moreover, \( W \) contains \( {Q}^{ * } \) as a subpath. The two edges at \( {x}^{ * } \) of this path lie in \( G \) ; in this sense, \( W \) already satisfies \( \left( *\right) \) . Let us now modify \( W \) so that \( W \) passes through every vertex of \( C \) exactly once. Simultaneously, we shall prepare for the later inclusion of the paths from \( {\mathcal{P}}_{1} \) by defining a map \( v \mapsto e\left( v\right) \) that is injective on \( {X}_{1} \) \( e\left( v\right) \) and assigns to every \( v \in {X}_{1} \) an edge \( e\left( v\right) \) of the modified \( W \) with the following property: The edge \( e\left( v\right) \) either bridges \( v \) or is incident with it. In the \( \left( {* * }\right) \) latter case, \( e\left( v\right) \in C \) and \( e\left( v\right) \neq v{x}^{ * } \) . For simplicity, we shall define the map \( v \mapsto e\left( v\right) \) on all of \( V\left( C\right) \smallsetminus {X}_{2} \) , a set that includes \( {X}_{1} \) by (2). To ensure injectivity on \( {X}_{1} \), we only have to make sure that no edge \( {vw} \in C \) is chosen both as \( e\left( v\right) \) and as \( e\left( w\right) \) . Indeed, since \( \left| {X}_{1}\right| \geq 2 \) if injectivity is a problem, and \( {\mathcal{P}}_{2} \neq \varnothing \) by assumption, we have \( \left| {C - {y}^{ * }}\right| \geq \left| {X}_{1}\right| + 2\left| {\mathcal{P}}_{2}\right| \geq 4 \) and hence \( \left| C\right| \geq 5 \) ; thus, no edge of \( {G}^{2} \) can bridge more than one vertex of \( C \), or bridge a vertex of \( C \) and lie on \( C \) at the same time. For our intended adjustments of \( W \) at the vertices of \( C \), we consider the intervals of \( C \) one at a time. By definition of \( W \), every interval is of one of the following three types: Type 1: \( W \) traverses \( I \) once; Type 2: \( W \) traverses \( I \) twice, in one direction and back immediately afterwards (formally: \( W \) contains a triple \( \left( {e, x, e}\right) \) with \( x \in {X}_{2} \) and \( e \in E\left( I\right) \) ); Type 3: \( W \) traverses \( I \) twice, on separate occasions (i.e., there is no triple as above). By definition of \( W \), the interval \( {I}^{ * } \) is of type 1 . The vertex \( x \) in the dead end definition of a type 2 interval will be called the dead end of that interval. Finally, since \( {Q}^{ * } \) is a subpath of \( W \) and \( W \) traverses both \( {I}^{ * } \) and \( {P}^{ * } \) only once, we have: The interval to the right of \( {I}^{ * } \) is of type 2 and has its dead (4) end on the left. --- \( I,{x}_{1},{x}_{2} \) \( {y}_{1},{y}_{2} \) \( {I}^{ - } \) --- Consider a fixed interval \( I = \left\lbrack {{x}_{1},{x}_{2}}\right\rbrack \) . Let \( {y}_{1} \) be the neighbour of \( {x}_{1} \), and \( {y}_{2} \) the neighbour of \( {x}_{2} \) on a path in \( {\m

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

averses \( I \) twice, in one direction and back immediately afterwards (formally: \( W \) contains a triple \( \left( {e, x, e}\right) \) with \( x \in {X}_{2} \) and \( e \in E\left( I\right) \) ); Type 3: \( W \) traverses \( I \) twice, on separate occasions (i.e., there is no triple as above). By definition of \( W \), the interval \( {I}^{ * } \) is of type 1 . The vertex \( x \) in the dead end definition of a type 2 interval will be called the dead end of that interval. Finally, since \( {Q}^{ * } \) is a subpath of \( W \) and \( W \) traverses both \( {I}^{ * } \) and \( {P}^{ * } \) only once, we have: The interval to the right of \( {I}^{ * } \) is of type 2 and has its dead (4) end on the left. --- \( I,{x}_{1},{x}_{2} \) \( {y}_{1},{y}_{2} \) \( {I}^{ - } \) --- Consider a fixed interval \( I = \left\lbrack {{x}_{1},{x}_{2}}\right\rbrack \) . Let \( {y}_{1} \) be the neighbour of \( {x}_{1} \), and \( {y}_{2} \) the neighbour of \( {x}_{2} \) on a path in \( {\mathcal{P}}_{2} \) . Let \( {I}^{ - } \) denote the interval to the left of \( I \) . Suppose first that \( I \) is of type 1 . We then leave \( W \) unchanged on \( I \) . If \( I \neq {I}^{ * } \) we choose as \( e\left( v\right) \), for each \( v \in I \), the edge to the left of \( v \) . As \( {I}^{ - } \neq {I}^{ * } \) by (4), and hence \( {x}_{1} \neq {x}^{ * } \), these choices of \( e\left( v\right) \) satisfy \( \left( {* * }\right) \) . If \( I = {I}^{ * } \), we define \( e\left( v\right) \) as the edge left of \( v \) if \( v \in \left( {{x}_{1},{x}^{ * }}\right\rbrack \cap I \), and as the edge right of \( v \) if \( v \in \left( {{x}^{ * },{x}_{2}}\right) \) . These choices of \( e\left( v\right) \) are again compatible with \( \left( {* * }\right) \) . Suppose now that \( I \) is of type 2. Assume first that \( {x}_{2} \) is the dead end of \( I \) . Then \( W \) contains the walk \( {y}_{1}{x}_{1}I{x}_{2}I{x}_{1}{I}^{ - } \) (possibly in reverse order). We now apply Lemma 10.3.2 (i) with \( P \mathrel{\text{:=}} {y}_{1}{x}_{1}I{\mathring{x}}_{2} \), and replace in \( W \) the subwalk \( {y}_{1}{x}_{1}I{x}_{2}I{x}_{1} \) by the \( {y}_{1} - {x}_{1} \) path \( Q \subseteq {G}^{2} \) of the lemma (Fig. 10.3.5). Then \( V\left( Q\right) = V\left( P\right) \smallsetminus \left\{ {{y}_{1},{x}_{1}}\right\} = V\left( I\right) \) . The vertices  Fig. 10.3.5. How to modify \( W \) on an interval of type 2 \( v \in \left( {{x}_{1},{x}_{2}^{ - }}\right) \) are each bridged by an edge of \( Q \), which we choose as \( e\left( v\right) \) . As \( e\left( {x}_{2}^{ - }\right) \) we choose the edge to the left of \( {x}_{2}^{ - } \) (unless \( {x}_{2}^{ - } = {x}_{1} \) ). This edge, too, lies on \( Q \), by the lemma. Moreover, by (4) it is not incident with \( {x}^{ * } \) (since \( {x}_{2} \) is the dead end of \( I \), by assumption) and hence satisfies \( \left( {* * }\right) \) . The case that \( {x}_{1} \) is the dead end of \( I \) can be treated in the same way: using Lemma 10.3.2 (i), we replace in \( W \) the subwalk \( {y}_{2}{x}_{2}I{x}_{1}I{x}_{2} \) by a \( {y}_{2} - {x}_{2} \) path \( Q \subseteq {G}^{2} \) with \( V\left( Q\right) = V\left( I\right) \), choose as \( e\left( v\right) \) for \( v \in \left( {{x}_{1}^{ + },{x}_{2}}\right) \) an edge of \( Q \) bridging \( v \), and define \( e\left( {x}_{1}^{ + }\right) \) as the edge to the right of \( {x}_{1}^{ + } \) (unless \( {x}_{1}^{ + } = {x}_{2} \) ). Suppose finally that \( I \) is of type 3. Since \( W \) traverses the edge \( {y}_{1}{x}_{1} \) only once and the interval \( {I}^{ - } \) no more than twice, \( W \) contains \( {y}_{1}{x}_{1}I \) and \( {I}^{ - } \cup I \) as subpaths, and \( {I}^{ - } \) is of type 1. By (4), however, \( {I}^{ - } \neq {I}^{ * } \) . Hence, when \( e\left( v\right) \) was defined for the vertices \( v \in {I}^{ - } \), the rightmost edge \( {x}_{1}^{ - }{x}_{1} \) of \( {I}^{ - } \) was not chosen as \( e\left( v\right) \) for any \( v \), so we may now replace this edge. Since \( W \) traverses \( {I}^{ + } \) no more than twice, it must traverse the edge \( {x}_{2}{y}_{2} \) immediately after one of its two subpaths \( {y}_{1}{x}_{1}I \) and \( {x}_{1}^{ - }{x}_{1}I \) . Take the starting vertex of this subpath \( \left( {y}_{1}\right. \) or \( \left. {x}_{1}^{ - }\right) \) as the vertex \( u \) in Lemma 10.3.2 (ii), and the other vertex in \( \left\{ {{y}_{1},{x}_{1}^{ - }}\right\} \) as \( {v}_{0} \) ; moreover, set \( {v}_{k} \mathrel{\text{:=}} {x}_{2} \) and \( w \mathrel{\text{:=}} {y}_{2} \) . Then the lemma enables us to replace these two subpaths of \( W \) between \( \left\{ {{y}_{1},{x}_{1}^{ - }}\right\} \) and \( \left\{ {{x}_{2},{y}_{2}}\right\} \) by disjoint paths in \( {G}^{2} \) (Fig. 10.3.6), and furthermore assigns to every vertex \( v \in I \) an edge \( e\left( v\right) \) of one of those paths, bridging \( v \) .  Fig. 10.3.6. A type 3 modification for the case \( u = {y}_{1} \) and \( k \) odd Following the above modifications, \( W \) is now a closed walk in \( {\widetilde{G}}^{2} \) . Let us check that, moreover, \( W \) contains every vertex of \( \widetilde{G} \) exactly once. For vertices of the paths in \( {\mathcal{P}}_{2} \) this is clear, because \( W \) still traverses every such path once and avoids it otherwise. For the vertices of \( C - {X}_{2} \), it follows from the above modifications by Lemma 10.3.2. So how about the vertices in \( {X}_{2} \) ? Let \( x \in {X}_{2} \) be given, and let \( y \) be its neighbour on a path in \( {\mathcal{P}}_{2} \) . Let \( {I}_{1} \) denote the interval \( I \) that satisfied \( {yxI} \subseteq W \) before the modification of \( W \), and let \( {I}_{2} \) denote the other interval ending in \( x \) . If \( {I}_{1} \) is of type 1, then \( {I}_{2} \) is of type 2 with dead end \( x \) . In this case, \( x \) was retained in \( W \) when \( W \) was modified on \( {I}_{1} \) but skipped when \( W \) was modified on \( {I}_{2} \) , and is thus contained exactly once in \( W \) now. If \( {I}_{1} \) is of type 2, then \( x \) is not its dead end, and \( {I}_{2} \) is of type 1 . The subwalk of \( W \) that started with \( {yx} \) and then went along \( {I}_{1} \) and back, was replaced with a \( y - x \) path. This path is now followed on \( W \) by the unchanged interval \( {I}_{2} \), so in this case too the vertex \( x \) is now contained in \( W \) exactly once. Finally, if \( {I}_{1} \) is of type 3, then \( x \) was contained in one of the replacement paths \( Q,{Q}^{\prime } \) from Lemma 10.3.2 (ii); as these paths were disjoint by the assertion of the lemma, \( x \) is once more left on \( W \) exactly once. We have thus shown that \( W \), after the modifications, is a closed walk in \( {\widetilde{G}}^{2} \) containing every vertex of \( \widetilde{G} \) exactly once, so \( W \) defines a Hamilton cycle \( \widetilde{H} \) of \( {\widetilde{G}}^{2} \) . Since \( W \) still contains the path \( {Q}^{ * },\widetilde{H} \) satisfies \( \left( *\right) \) . Up until now, we have assumed that \( {\mathcal{P}}_{2} \) is non-empty. If \( {\mathcal{P}}_{2} = \varnothing \) , let us set \( \widetilde{H} \mathrel{\text{:=}} \widetilde{G} = C \) ; then, again, \( \widetilde{H} \) satisfies \( \left( *\right) \) . It remains to turn \( \widetilde{H} \) into a Hamilton cycle \( H \) of \( {G}^{2} \) by incorporating the paths from \( {\mathcal{P}}_{1} \) . In order to be able to treat the case of \( {\mathcal{P}}_{2} = \varnothing \) along with the case of \( {\mathcal{P}}_{2} \neq \varnothing \), we define a map \( v \mapsto e\left( v\right) \) also when \( {\mathcal{P}}_{2} = \varnothing \), as follows: for every \( v \in C - {y}^{ * } \), set \[ e\left( v\right) \mathrel{\text{:=}} \left\{ \begin{array}{ll} v{v}^{ + } & \text{if }v \in \left\lbrack {{x}^{ * },{y}^{ * }}\right) \\ v{v}^{ - } & \text{if }v \in \left( {{y}^{ * },{x}^{ * }}\right) . \end{array}\right. \] (Here, \( \left\lbrack {{x}^{ * },{y}^{ * }}\right) \) and \( \left( {{y}^{ * },{x}^{ * }}\right) \) denote the obvious paths in \( C \) defined analogously to intervals.) As before, this map \( v \mapsto e\left( v\right) \) is injective, satisfies \( \left( {* * }\right) \), and is defined on a superset of \( {X}_{1} \) ; recall that \( {y}^{ * } \) cannot lie in \( {X}_{1} \) by definition. \( P, v \) Let \( P \in {\mathcal{P}}_{1} \) be a path to be incorporated into \( \widetilde{H} \), say with foot \( {y}_{1},{y}_{2} \) \( v \in {X}_{1} \) and ends \( {y}_{1},{y}_{2} \) . (If \( \left| P\right| = 1 \), then \( {y}_{1} = {y}_{2} \) .) Our aim is to replace \( e \) the edge \( e \mathrel{\text{:=}} e\left( v\right) \) in \( \widetilde{H} \) by \( P \) ; we thus have to show that the ends of \( P \) are joined to those of \( e \) by suitable edges of \( {G}^{2} \) . By (2) and (3), \( v \) has only two neighbours in \( \widetilde{G} \), its neighbours \( {x}_{1},{x}_{2} \) on \( C \) . If \( v \) is incident with \( e \), i.e. if \( e = v{x}_{i} \) with \( i \in \{ 1,2\} \), we replace \( e \) by the path \( v{y}_{1}P{y}_{2}{x}_{i} \subseteq {G}^{2} \) (Fig. 10.3.7). If \( v \) is not incident  Fig. 10.3.7. Replacing the edge \( e \) in \( \widetilde{H} \) with \( e \) then \( e \) bridges \( v \), by \( \left( {* * }\right) \) . Then \( e = {x}_{1}{x}_{2} \), and we replace \( e \) by the path \( {x}_{1}{y}_{1}P{y}_{2}{x}_{2} \subseteq {G}^{2} \) (Fig. 10.3.8). Since \( v \mapsto e\left( v\right) \) is injective on \( {X}_{1} \), assertion (2) implies that all these modifications of \( \widetilde{H} \) (one for every \( P \in {\mathcal{P}}_{1} \) ) can be

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

he ends of \( P \) are joined to those of \( e \) by suitable edges of \( {G}^{2} \) . By (2) and (3), \( v \) has only two neighbours in \( \widetilde{G} \), its neighbours \( {x}_{1},{x}_{2} \) on \( C \) . If \( v \) is incident with \( e \), i.e. if \( e = v{x}_{i} \) with \( i \in \{ 1,2\} \), we replace \( e \) by the path \( v{y}_{1}P{y}_{2}{x}_{i} \subseteq {G}^{2} \) (Fig. 10.3.7). If \( v \) is not incident  Fig. 10.3.7. Replacing the edge \( e \) in \( \widetilde{H} \) with \( e \) then \( e \) bridges \( v \), by \( \left( {* * }\right) \) . Then \( e = {x}_{1}{x}_{2} \), and we replace \( e \) by the path \( {x}_{1}{y}_{1}P{y}_{2}{x}_{2} \subseteq {G}^{2} \) (Fig. 10.3.8). Since \( v \mapsto e\left( v\right) \) is injective on \( {X}_{1} \), assertion (2) implies that all these modifications of \( \widetilde{H} \) (one for every \( P \in {\mathcal{P}}_{1} \) ) can be performed independently, and hence produce a Hamilton cycle \( H \) of \( {G}^{2} \) .  Fig. 10.3.8. Replacing the edge \( e \) in \( \widetilde{H} \) Let us finally check that \( H \) satisfies \( \left( *\right) \), i.e. that both edges of \( H \) at \( {x}^{ * } \) lie in \( G \) . Since \( \left( *\right) \) holds for \( \widetilde{H} \), it suffices to show that any edge \( e = {x}^{ * }z \) of \( \widetilde{H} \) that is not in \( H \) (and hence has the form \( e = e\left( v\right) \) for some \( v \in {X}_{1} \) ) was replaced by an \( {x}^{ * } - z \) path whose first edge lies in \( G \) . Where can the vertex \( v \) lie? Let us show that \( v \) must be incident with \( e \) . If not then \( {\mathcal{P}}_{2} \neq \varnothing \), and \( e \) bridges \( v \) . Now \( {\mathcal{P}}_{2} \neq \varnothing \) and \( v \in {X}_{1} \) together imply that \( \left| {C - {y}^{ * }}\right| \geq \left| {X}_{1}\right| + 2\left| {\mathcal{P}}_{2}\right| \geq 3 \), so \( \left| C\right| \geq 4 \) . As \( e \in G \) (by \( \left( *\right) \) for \( \widetilde{H} \) ), the fact that \( e \) bridges \( v \) thus contradicts (3). So \( v \) is indeed incident with \( e \) . Hence \( v \in \left\{ {{x}^{ * }, z}\right\} \) by definition of \( e \) , while \( e \neq v{x}^{ * } \) by \( \left( {* * }\right) \) . Thus \( v = {x}^{ * } \), and \( e \) was replaced by a path of the form \( {x}^{ * }{y}_{1}P{y}_{2}z \) . Since \( {x}^{ * }{y}_{1} \) is an edge of \( G \), this replacement again preserves \( \left( *\right) \) . Therefore \( H \) does indeed satisfy \( \left( *\right) \), and our induction is complete. Just like Tutte's theorem (10.1.3), Fleischner's theorem might extend to infinite graphs with circles: Conjecture. The square of every 2-connected locally finite graph contains a Hamilton circle. We close the chapter with a far-reaching conjecture generalizing Dirac's theorem: Conjecture. (Seymour 1974) Let \( G \) be a graph of order \( n \geq 3 \), and let \( k \) be a positive integer. If \( G \) has minimum degree \[ \delta \left( G\right) \geq \frac{k}{k + 1}n \] then \( G \) has a Hamilton cycle \( H \) such that \( {H}^{k} \subseteq G \) . For \( k = 1 \), this is precisely Dirac’s theorem. The conjecture was proved for large enough \( n \) (depending on \( k \) ) by Komlós, Sárközy and Szemerédi (1998). ## Exercises 1. An oriented complete graph is called a tournament. Show that every tournament contains a (directed) Hamilton path. 2. Show that every uniquely 3-edge-colourable cubic graph is hamiltonian. ('Unique' means that all 3-edge-colourings induce the same edge partition.) 3. Given an even positive integer \( k \), construct for every \( n \geq k \) a \( k \) -regular graph of order \( {2n} + 1 \) . 4. \( {}^{ - } \) Prove or disprove the following strengthening of Proposition 10.1.2: ’Every \( k \) -connected graph \( G \) with \( \left| G\right| \geq 3 \) and \( \chi \left( G\right) \geq \left| G\right| /k \) has a Hamilton cycle.' 5. \( {\left( \mathrm{i}\right) }^{ - } \) Show that hamiltonian graphs are 1-tough. (ii) Find a graph that is 1-tough but not hamiltonian. 6. Prove the toughness conjecture for planar graphs. Does it hold with \( t = 2 \), or even with some \( t < 2 \) ? 7. \( {}^{ - } \) Find a hamiltonian graph whose degree sequence is not hamiltonian. 8. \( {}^{ - } \) Let \( G \) be a graph with fewer than \( i \) vertices of degree at most \( i \), for every \( i < \left| G\right| /2 \) . Use Chvátal’s theorem to show that \( G \) is hamiltonian. (Thus in particular, Chvátal's theorem implies Dirac's theorem.) 9. Prove that the square \( {G}^{2} \) of a \( k \) -connected graph \( G \) is \( k \) -tough. Use this to deduce Fleischner's theorem for graphs satisfying the toughness conjecture with \( t = 2 \) . 10. Show that Exercise 5(i) has the following weak converse: for every non-hamiltonian graph \( G \) there exists a graph \( {G}^{\prime } \) that has a pointwise greater degree-sequence than \( G \) but is not 1-tough. 11. Find a connected graph \( G \) whose square \( {G}^{2} \) has no Hamilton cycle. 12. \( {}^{ + } \) Show by induction on \( \left| G\right| \) that the third power \( {G}^{3} \) of a connected graph \( G \) contains a Hamilton path between any two vertices. Deduce that \( {G}^{3} \) is hamiltonian. 13. \( {}^{ + } \) Let \( G \) be a graph in which every vertex has odd degree. Show that every edge of \( G \) lies on an even number of Hamilton cycles. (Hint. Let \( {xy} \in E\left( G\right) \) be given. The Hamilton cycles through \( {xy} \) correspond to the Hamilton paths in \( G - {xy} \) from \( x \) to \( y \) . Consider the set \( \mathcal{H} \) of all Hamilton paths in \( G - {xy} \) starting at \( x \), and show that an even number of these end in \( y \) . To show this, define a graph on \( \mathcal{H} \) so that the desired assertion follows from Proposition 1.2.1.) ## Notes The problem of finding a Hamilton cycle in a graph has the same kind of origin as its Euler tour counterpart and the four colour problem: all three problems come from mathematical puzzles older than graph theory itself. What began as a game invented by W.R.Hamilton in 1857-in which 'Hamilton cycles' had to be found on the graph of the dodecahedron-reemerged over a hundred years later as a combinatorial optimization problem of prime importance: the travelling salesman problem. Here, a salesman has to visit a number of customers, and his problem is to arrange these in a suitable circular route. (For reasons not included in the mathematical brief, the route has to be such that after visiting a customer the salesman does not pass through that town again.) Much of the motivation for considering Hamilton cycles comes from variations of this algorithmic problem. The lack of a good characterization of hamiltonicity also has to do with an algorithmic problem: deciding whether or not a given graph is hamiltonian is NP-hard (indeed, this was one of the early prototypes of an NP-complete decision problem), while the existence of a good characterization would place it in \( \mathrm{{NP}} \cap \mathrm{{co}} - \mathrm{{NP}} \), which is widely believed to equal P. Thus, unless \( \mathrm{P} = \mathrm{{NP}} \), no good characterization of hamiltonicity exists. See the introduction to Chapter 12.5 , or the end of the notes for Chapter 12, for more. The 'proof' of the four colour theorem indicated at the end of Section 10.1, which is based on the (false) premise that every 3-connected cubic planar graph is hamiltonian, is usually attributed to the Scottish mathematician P.G. Tait. Following Kempe's flawed proof of 1879 (see the notes for Chapter 5), it seems that Tait believed to be in possession of at least one 'new proof of Kempe's theorem'. However, when he addressed the Edinburgh Mathematical Society on this subject in 1883, he seems to have been aware that he could not-really-prove the above statement about Hamilton cycles. His account in P.G. Tait, Listing's topologie, Phil. Mag. 17 (1884), 30-46, makes some entertaining reading. A shorter proof of Tutte's theorem that 4-connected planar graphs are hamiltonian has been given by C. Thomassen, A theorem on paths in planar graphs, J. Graph Theory 7 (1983), 169-176. Tutte's counterexample to Tait's assumption that even 3-connectedness suffices (at least for cubic graphs) is shown in Bollobás, and in J.A.Bondy & U.S.R. Murty, Graph Theory with Applications, Macmillan 1976 (where Tait's attempted proof is discussed in some detail). Bruhn's conjecture generalizing Tutte's theorem to infinite graphs was first stated in R. Diestel, The cycle space of an infinite graph, Combinatorics, Probability and Computing 14 (2005), 59-79. As the notion of a Hamilton circle is relatively recent, earlier generalizations of Hamilton cycle theorems asked for spanning double rays. Now a ray can pass through a finite separator only finitely often, so a necessary condition for the existence of a spanning ray or double ray is that the graph has at most one or two ends, respectively. Confirming a long-standing conjecture of Nash-Williams, X. Yu, Infinite paths in planar graphs I-III (preprints 2004) announced that a 4-connected planar graph with at most two ends contains a spanning double ray. N. Dean, R. Thomas and X Yu, Spanning paths in infinite planar graphs, J. Graph Theory 23 (1996), 163-174, proved Nash-Williams's conjecture that a one-ended 4-connected planar graph has a spanning ray. Proposition 10.1.2 is due to Chvátal and Erdős (1972). The toughness invariant and conjecture were proposed by V. Chvátal, Tough graphs and hamiltonian circuits, Discrete Math. 5 (1973),215-228. If true with \( t = 2 \), the conjecture would have implied Fleischner's thereom; see Exercise 9. However, it was disproved for \( t = 2 \) by D. Bauer, H.J. Broersm

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

nning double rays. Now a ray can pass through a finite separator only finitely often, so a necessary condition for the existence of a spanning ray or double ray is that the graph has at most one or two ends, respectively. Confirming a long-standing conjecture of Nash-Williams, X. Yu, Infinite paths in planar graphs I-III (preprints 2004) announced that a 4-connected planar graph with at most two ends contains a spanning double ray. N. Dean, R. Thomas and X Yu, Spanning paths in infinite planar graphs, J. Graph Theory 23 (1996), 163-174, proved Nash-Williams's conjecture that a one-ended 4-connected planar graph has a spanning ray. Proposition 10.1.2 is due to Chvátal and Erdős (1972). The toughness invariant and conjecture were proposed by V. Chvátal, Tough graphs and hamiltonian circuits, Discrete Math. 5 (1973),215-228. If true with \( t = 2 \), the conjecture would have implied Fleischner's thereom; see Exercise 9. However, it was disproved for \( t = 2 \) by D. Bauer, H.J. Broersma &H.J. Veldman, Not every 2-tough graph is hamiltonian, Discrete Appl. Math. 99 (2000), 317- 321. Theorem 10.2.1 is due to V. Chvátal, On Hamilton's ideals, J. Combin. Theory B 12 (1972), 163–168. Our proof of Fleischner's theorem is based on S. Ríha, A new proof of the theorem by Fleischner, J. Combin. Theory B 52 (1991), 117-123. C. Thomas-sen, Hamiltonian paths in squares of infinite locally finite blocks, Ann. Discrete Math. 3 (1978), 269-277, proved that the square of every 2-connected one-ended locally finite graph contains a spanning ray. Seymour's conjecture is from P.D. Seymour, Problem 3, in (T.P. McDonough and V.C. Mavron, eds.) Combinatorics, Cambridge University Press 1974. Its proof for large \( n \) is due to J. Komlós, G.N. Sárközy &E. Szemerédi, Proof of the Seymour conjecture for large graphs, Ann. Comb. 2 (1998), 43-60. 11 ## Random Graphs At various points in this book, we already encountered the following fundamental theorem of Erdős: for every integer \( k \) there is a graph \( G \) with \( g\left( G\right) > k \) and \( \chi \left( G\right) > k \) . In plain English: there exist graphs combining arbitrarily large girth with arbitrarily high chromatic number. How could one prove such a theorem? The standard approach would be to construct a graph with those two properties, possibly in steps by induction on \( k \) . However, this is anything but straightforward: the global nature of the second property forced by the first, namely, that the graph should have high chromatic number 'overall' but be acyclic (and hence 2-colourable) locally, flies in the face of any attempt to build it up, constructively, from smaller pieces that have the same or similar properties. In his pioneering paper of 1959, Erdős took a radically different approach: for each \( n \) he defined a probability space on the set of graphs with \( n \) vertices, and showed that, for some carefully chosen probability measures, the probability that an \( n \) -vertex graph has both of the above properties is positive for all large enough \( n \) . This approach, now called the probabilistic method, has since unfolded into a sophisticated and versatile proof technique, in graph theory as much as in other branches of discrete mathematics. The theory of random graphs is now a subject in its own right. The aim of this chapter is to offer an elementary but rigorous introduction to random graphs: no more than is necessary to understand its basic concepts, ideas and techniques, but enough to give an inkling of the power and elegance hidden behind the calculations. Erdős's theorem asserts the existence of a graph with certain properties: it is a perfectly ordinary assertion showing no trace of the randomness employed in its proof. There are also results in random graphs that are generically random even in their statement: these are theorems about almost all graphs, a notion we shall meet in Section 11.3. In the last section, we give a detailed proof of a theorem of Erdős and Rényi that illustrates a proof technique frequently used in random graphs, the so-called second moment method. ## 11.1 The notion of a random graph Let \( V \) be a fixed set of \( n \) elements, say \( V = \{ 0,\ldots, n - 1\} \) . Our aim is to turn the set \( \mathcal{G} \) of all graphs on \( V \) into a probability space, and then to consider the kind of questions typically asked about random objects: What is the probability that a graph \( G \in \mathcal{G} \) has this or that property? What is the expected value of a given invariant on \( G \), say its expected girth or chromatic number? Intuitively, we should be able to generate \( G \) randomly as follows. For each \( e \in {\left\lbrack V\right\rbrack }^{2} \) we decide by some random experiment whether or not \( e \) shall be an edge of \( G \) ; these experiments are performed independently, and for each the probability of success-i.e. of accepting \( e \) as an edge for \( G \) -is equal to some fixed \( {}^{1} \) number \( p \in \left\lbrack {0,1}\right\rbrack \) . Then if \( {G}_{0} \) is some fixed graph on \( V \), with \( m \) edges say, the elementary event \( \left\{ {G}_{0}\right\} \) has a probability of \( {p}^{m}{q}^{\left( \begin{matrix} n \\ 2 \end{matrix}\right) - m} \) (where \( q \mathrel{\text{:=}} 1 - p \) ): with this probability, our randomly generated graph \( G \) is this particular graph \( {G}_{0} \) . (The probability that \( G \) is isomorphic to \( {G}_{0} \) will usually be greater.) But if the probabilities of all the elementary events are thus determined, then so is the entire probability measure of our desired space \( \mathcal{G} \) . Hence all that remains to be checked is that such a probability measure on \( \mathcal{G} \), one for which all individual edges occur independently with probability \( p \), does indeed exist. \( {}^{2} \) In order to construct such a measure on \( \mathcal{G} \) formally, we start by defining for every potential edge \( e \in {\left\lbrack V\right\rbrack }^{2} \) its own little probability space \( {\Omega }_{e} \) \( {\Omega }_{e} \mathrel{\text{:=}} \left\{ {{0}_{e},{1}_{e}}\right\} \), choosing \( {P}_{e}\left( \left\{ {1}_{e}\right\} \right) \mathrel{\text{:=}} p \) and \( {P}_{e}\left( \left\{ {0}_{e}\right\} \right) \mathrel{\text{:=}} q \) as the \( {P}_{e} \) probabilities of its two elementary events. As our desired probability \( \mathcal{G}\left( {n, p}\right) \) space \( \mathcal{G} = \mathcal{G}\left( {n, p}\right) \) we then take the product space \[ \Omega \mathrel{\text{:=}} \mathop{\prod }\limits_{{e \in {\left\lbrack V\right\rbrack }^{2}}}{\Omega }_{e} \] --- 1 Often, the value of \( p \) will depend on the cardinality \( n \) of the set \( V \) on which our random graphs are generated; thus, \( p \) will be the value \( p = p\left( n\right) \) of some function \( n \mapsto p\left( n\right) \) . Note, however, that \( V \) (and hence \( n \) ) is fixed for the definition of \( \mathcal{G} \) : for each \( n \) separately, we are constructing a probability space of the graphs \( G \) on \( V = \{ 0,\ldots, n - 1\} \), and within each space the probability that \( e \in {\left\lbrack V\right\rbrack }^{2} \) is an edge of \( G \) has the same value for all \( e \) . 2 Any reader ready to believe this may skip ahead now to the end of Proposition 11.1.1, without missing anything. --- Thus, formally, an element of \( \Omega \) is a map \( \omega \) assigning to every \( e \in {\left\lbrack V\right\rbrack }^{2} \) either \( {0}_{e} \) or \( {1}_{e} \), and the probability measure \( P \) on \( \Omega \) is the product measure of all the measures \( {P}_{e} \) . In practice, of course, we identify \( \omega \) with the graph \( G \) on \( V \) whose edge set is \[ E\left( G\right) = \left\{ {e \mid \omega \left( e\right) = {1}_{e}}\right\} \] and call \( G \) a random graph on \( V \) with edge probability \( p \) . --- random graph --- Following standard probabilistic terminology, we may now call any set of graphs on \( V \) an event in \( \mathcal{G}\left( {n, p}\right) \) . In particular, for every \( e \in {\left\lbrack V\right\rbrack }^{2} \) event the set \[ {A}_{e} \mathrel{\text{:=}} \left\{ {\omega \mid \omega \left( e\right) = {1}_{e}}\right\} \] of all graphs \( G \) on \( V \) with \( e \in E\left( G\right) \) is an event: the event that \( e \) is an edge of \( G \) . For these events, we can now prove formally what had been our guiding intuition all along: Proposition 11.1.1. The events \( {A}_{e} \) are independent and occur with probability \( p \) . Proof. By definition, \[ {A}_{e} = \left\{ {1}_{e}\right\} \times \mathop{\prod }\limits_{{{e}^{\prime } \neq e}}{\Omega }_{{e}^{\prime }} \] Since \( P \) is the product measure of all the measures \( {P}_{e} \), this implies \[ P\left( {A}_{e}\right) = p \cdot \mathop{\prod }\limits_{{{e}^{\prime } \neq e}}1 = p. \] Similarly, if \( \left\{ {{e}_{1},\ldots ,{e}_{k}}\right\} \) is any subset of \( {\left\lbrack V\right\rbrack }^{2} \), then \[ \begin{aligned} P\left( {{A}_{{e}_{1}} \cap \ldots \cap {A}_{{e}_{k}}}\right) & = P\left( {\left\{ {1}_{{e}_{1}}\right\} \times \ldots \times \left\{ {1}_{{e}_{k}}\right\} \times \mathop{\prod }\limits_{{e \notin \left\{ {{e}_{1},\ldots ,{e}_{k}}\right\} }}{\Omega }_{e}}\right) \\ & = {p}^{k} \end{aligned} \] \[ = P\left( {A}_{{e}_{1}}\right) \cdots P\left( {A}_{{e}_{k}}\right) . \] As noted before, \( P \) is determined uniquely by the value of \( p \) and our assumption that the events \( {A}_{e} \) are independent. In order to calculate probabilities in \( \mathcal{G}\left( {n, p}\right) \), it therefore generally suffices to work with these two assumptions: our concrete model for \( \mathcal{G}\left( {n, p}\right) \) has served its purpose and will not be needed again. As a simple example of such a calculation, consider the event that \( G \) contains some fixed graph \( H \) on a subset of \( V \) as

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

right\} \) is any subset of \( {\left\lbrack V\right\rbrack }^{2} \), then \[ \begin{aligned} P\left( {{A}_{{e}_{1}} \cap \ldots \cap {A}_{{e}_{k}}}\right) & = P\left( {\left\{ {1}_{{e}_{1}}\right\} \times \ldots \times \left\{ {1}_{{e}_{k}}\right\} \times \mathop{\prod }\limits_{{e \notin \left\{ {{e}_{1},\ldots ,{e}_{k}}\right\} }}{\Omega }_{e}}\right) \\ & = {p}^{k} \end{aligned} \] \[ = P\left( {A}_{{e}_{1}}\right) \cdots P\left( {A}_{{e}_{k}}\right) . \] As noted before, \( P \) is determined uniquely by the value of \( p \) and our assumption that the events \( {A}_{e} \) are independent. In order to calculate probabilities in \( \mathcal{G}\left( {n, p}\right) \), it therefore generally suffices to work with these two assumptions: our concrete model for \( \mathcal{G}\left( {n, p}\right) \) has served its purpose and will not be needed again. As a simple example of such a calculation, consider the event that \( G \) contains some fixed graph \( H \) on a subset of \( V \) as a subgraph; let \( \left| H\right| = : k \) and \( \parallel H\parallel = : \ell \) . The probability of this event \( H \subseteq G \) is the product of the probabilities \( {A}_{e} \) over all the edges \( e \in H \), so \( P\left\lbrack {H \subseteq G}\right\rbrack = {p}^{\ell } \) . In contrast, the probability that \( H \) is an induced subgraph of \( G \) is \( {p}^{\ell }{q}^{\left( \begin{matrix} k \\ 2 \end{matrix}\right) - \ell } \) : now the edges missing from \( H \) are required to be missing from \( G \) too, and they do so independently with probability \( q \) . The probability \( {P}_{H} \) that \( G \) has an induced subgraph isomorphic to \( H \) is usually more difficult to compute: since the possible instances of \( H \) on subsets of \( V \) overlap, the events that they occur in \( G \) are not independent. However, the sum (over all \( k \) -sets \( U \subseteq V \) ) of the probabilities \( P\left\lbrack {H \simeq G\left\lbrack U\right\rbrack }\right\rbrack \) is always an upper bound for \( {P}_{H} \), since \( {P}_{H} \) is the measure of the union of all those events. For example, if \( H = \overline{{K}^{k}} \), we have the following trivial upper bound on the probability that \( G \) contains an induced copy of \( H \) : Lemma 11.1.2. For all integers \( n, k \) with \( n \geq k \geq 2 \), the probability that \( G \in \mathcal{G}\left( {n, p}\right) \) has a set of \( k \) independent vertices is at most \[ P\left\lbrack {\alpha \left( G\right) \geq k}\right\rbrack \leq \left( \begin{array}{l} n \\ k \end{array}\right) {q}^{\left( \begin{array}{l} k \\ 2 \end{array}\right) }. \] Proof. The probability that a fixed \( k \) -set \( U \subseteq V \) is independent in \( G \) is \( {q}^{\left( \begin{matrix} k \\ 2 \end{matrix}\right) } \) . The assertion thus follows from the fact that there are only \( \left( \begin{array}{l} n \\ k \end{array}\right) \) such sets \( U \) . Analogously, the probability that \( G \in \mathcal{G}\left( {n, p}\right) \) contains a \( {K}^{k} \) is at most \[ P\left\lbrack {\omega \left( G\right) \geq k}\right\rbrack \leq \left( \begin{array}{l} n \\ k \end{array}\right) {p}^{\left( \begin{array}{l} k \\ 2 \end{array}\right) }. \] Now if \( k \) is fixed, and \( n \) is small enough that these bounds for the probabilities \( P\left\lbrack {\alpha \left( G\right) \geq k}\right\rbrack \) and \( P\left\lbrack {\omega \left( G\right) \geq k}\right\rbrack \) sum to less than 1, then \( \mathcal{G} \) contains graphs that have neither property: graphs which contain neither a \( {K}^{k} \) nor a \( \overline{{K}^{k}} \) induced. But then any such \( n \) is a lower bound for the Ramsey number of \( k \) ! As the following theorem shows, this lower bound is quite close to the upper bound of \( {2}^{{2k} - 3} \) implied by the proof of Theorem 9.1.1: Theorem 11.1.3. (Erdős 1947) For every integer \( k \geq 3 \), the Ramsey number of \( k \) satisfies \[ R\left( k\right) > {2}^{k/2}\text{.} \] Proof. For \( k = 3 \) we trivially have \( R\left( 3\right) \geq 3 > {2}^{3/2} \), so let \( k \geq 4 \) . We show that, for all \( n \leq {2}^{k/2} \) and \( G \in \mathcal{G}\left( {n,\frac{1}{2}}\right) \), the probabilities \( P\left\lbrack {\alpha \left( G\right) \geq k}\right\rbrack \) and \( P\left\lbrack {\omega \left( G\right) \geq k}\right\rbrack \) are both less than \( \frac{1}{2} \) . Since \( p = q = \frac{1}{2} \), Lemma 11.1.2 and the analogous assertion for \( \omega \left( G\right) \) imply the following for all \( n \leq {2}^{k/2} \) (use that \( k! > {2}^{k} \) for \( k \geq 4 \) ): \[ P\left\lbrack {\alpha \left( G\right) \geq k}\right\rbrack, P\left\lbrack {\omega \left( G\right) \geq k}\right\rbrack \leq \left( \begin{array}{l} n \\ k \end{array}\right) {\left( \frac{1}{2}\right) }^{\left( \begin{matrix} k \\ 2 \end{matrix}\right) } \] \[ < \left( {{n}^{k}/{2}^{k}}\right) {2}^{-\frac{1}{2}k\left( {k - 1}\right) } \] \[ \leq \left( {{2}^{{k}^{2}/2}/{2}^{k}}\right) {2}^{-\frac{1}{2}k\left( {k - 1}\right) } \] \[ = {2}^{-k/2} \] \[ < \frac{1}{2}\text{.} \] In the context of random graphs, each of the familiar graph invariants (like average degree, connectivity, girth, chromatic number, and so on) may be interpreted as a non-negative random variable on \( \mathcal{G}\left( {n, p}\right) \) , random variable a function \[ X : \mathcal{G}\left( {n, p}\right) \rightarrow \lbrack 0,\infty ). \] The mean or expected value of \( X \) is the number mean \[ E\left( X\right) \mathrel{\text{:=}} \mathop{\sum }\limits_{{G \in \mathcal{G}\left( {n, p}\right) }}P\left( {\{ G\} }\right) \cdot X\left( G\right) . \] expectation \( E\left( X\right) \) Note that the operator \( E \), the expectation, is linear: we have \( E\left( {X + Y}\right) = \) \( E\left( X\right) + E\left( Y\right) \) and \( E\left( {\lambda X}\right) = {\lambda E}\left( X\right) \) for any two random variables \( X, Y \) on \( \mathcal{G}\left( {n, p}\right) \) and \( \lambda \in \mathbb{R} \) . Computing the mean of a random variable \( X \) can be a simple and effective way to establish the existence of a graph \( G \) such that \( X\left( G\right) < a \) for some fixed \( a > 0 \) and, moreover, \( G \) has some desired property \( \mathcal{P} \) . Indeed, if the expected value of \( X \) is small, then \( X\left( G\right) \) cannot be large for more than a few graphs in \( \mathcal{G}\left( {n, p}\right) \), because \( X\left( G\right) \geq 0 \) for all \( G \in \mathcal{G}\left( {n, p}\right) \) . Hence \( X \) must be small for many graphs in \( \mathcal{G}\left( {n, p}\right) \), and it is reasonable to expect that among these we may find one with the desired property \( \mathcal{P} \) . This simple idea lies at the heart of countless non-constructive existence proofs using random graphs, including the proof of Erdős's theorem presented in the next section. Quantified, it takes the form of the following lemma, whose proof follows at once from the definition of the expectation and the additivity of \( P \) : Lemma 11.1.4. (Markov's Inequality) Let \( X \geq 0 \) be a random variable on \( \mathcal{G}\left( {n, p}\right) \) and \( a > 0 \) . Then \[ P\left\lbrack {X \geq a}\right\rbrack \leq E\left( X\right) /a. \] Proof. \[ E\left( X\right) = \mathop{\sum }\limits_{{G \in \mathcal{G}\left( {n, p}\right) }}P\left( {\{ G\} }\right) \cdot X\left( G\right) \] \[ \geq \mathop{\sum }\limits_{\substack{{G \in \mathcal{G}\left( {n, p}\right) } \\ {X\left( G\right) \geq a} }}P\left( {\{ G\} }\right) \cdot X\left( G\right) \] \[ \geq \mathop{\sum }\limits_{\substack{{G \in \mathcal{G}\left( {n, p}\right) } \\ {X\left( G\right) \geq a} }}P\left( {\{ G\} }\right) \cdot a \] \[ = P\left\lbrack {X \geq a}\right\rbrack \cdot a. \] Since our probability spaces are finite, the expectation can often be computed by a simple application of double counting, a standard combinatorial technique we met before in the proofs of Corollary 4.2.10 and Theorem 5.5.4. For example, if \( X \) is a random variable on \( \mathcal{G}\left( {n, p}\right) \) that counts the number of subgraphs of \( G \) in some fixed set \( \mathcal{H} \) of graphs on \( V \), then \( E\left( X\right) \), by definition, counts the number of pairs \( \left( {G, H}\right) \) such that \( H \in \mathcal{H} \) and \( H \subseteq G \), each weighted with the probability of \( \{ G\} \) . Algorithmically, we compute \( E\left( X\right) \) by going through the graphs \( G \in \) \( \mathcal{G}\left( {n, p}\right) \) in an ’outer loop’ and performing, for each \( G \), an ’inner loop’ that runs through the graphs \( H \in \mathcal{H} \) and counts ’ \( P\left( {\{ G\} }\right) \) ’ whenever \( H \subseteq G \) . Alternatively, we may count the same set of weighted pairs with \( H \) in the outer and \( G \) in the inner loop: this amounts to adding up, over all \( H \in \mathcal{H} \), the probabilities \( P\left\lbrack {H \subseteq G}\right\rbrack \) . To illustrate this once in detail, let us compute the expected number of cycles of some given length \( k \geq 3 \) in a random graph \( G \in \mathcal{G}\left( {n, p}\right) \) . So let \( X : \mathcal{G}\left( {n, p}\right) \rightarrow \mathbb{N} \) be the random variable that assigns to every random graph \( G \) its number of \( k \) -cycles, the number of subgraphs isomorphic to \( {C}^{k} \) . Let us write \( {\left( n\right) }_{k} \) \[ {\left( n\right) }_{k} \mathrel{\text{:=}} n\left( {n - 1}\right) \left( {n - 2}\right) \cdots \left( {n - k + 1}\right) \] for the number of sequences of \( k \) distinct elements of a given \( n \) -set. \( \left\lbrack \begin{array}{l} {11.2.2} \\ {11.4.3} \end{array}\right\rbrack \) Lemma 11.1.5. The expected number of \( k \) -cycles in \( G \in \mathcal{G}\left( {n, p}\right) \) is \[ E\left( X\right) = \frac{{\left( n\right) }_{k}}{2k}{p}^{k} \] Proof. For every \( k \) -cycle \( C \) with vertices in \( V = \{ 0,\ldots, n - 1\} \

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

To illustrate this once in detail, let us compute the expected number of cycles of some given length \( k \geq 3 \) in a random graph \( G \in \mathcal{G}\left( {n, p}\right) \) . So let \( X : \mathcal{G}\left( {n, p}\right) \rightarrow \mathbb{N} \) be the random variable that assigns to every random graph \( G \) its number of \( k \) -cycles, the number of subgraphs isomorphic to \( {C}^{k} \) . Let us write \( {\left( n\right) }_{k} \) \[ {\left( n\right) }_{k} \mathrel{\text{:=}} n\left( {n - 1}\right) \left( {n - 2}\right) \cdots \left( {n - k + 1}\right) \] for the number of sequences of \( k \) distinct elements of a given \( n \) -set. \( \left\lbrack \begin{array}{l} {11.2.2} \\ {11.4.3} \end{array}\right\rbrack \) Lemma 11.1.5. The expected number of \( k \) -cycles in \( G \in \mathcal{G}\left( {n, p}\right) \) is \[ E\left( X\right) = \frac{{\left( n\right) }_{k}}{2k}{p}^{k} \] Proof. For every \( k \) -cycle \( C \) with vertices in \( V = \{ 0,\ldots, n - 1\} \), the vertex set of the graphs in \( \mathcal{G}\left( {n, p}\right) \), let \( {X}_{C} : \mathcal{G}\left( {n, p}\right) \rightarrow \{ 0,1\} \) denote the indicator random variable of \( C \) : \[ {X}_{C} : G \mapsto \left\{ \begin{array}{ll} 1 & \text{ if }C \subseteq G \\ 0 & \text{ otherwise. } \end{array}\right. \] Since \( {X}_{C} \) takes only 1 as a positive value, its expectation \( E\left( {X}_{C}\right) \) equals the measure \( P\left\lbrack {{X}_{C} = 1}\right\rbrack \) of the set of all graphs in \( \mathcal{G}\left( {n, p}\right) \) that contain \( C \) . But this is just the probability that \( C \subseteq G \) : \[ E\left( {X}_{C}\right) = P\left\lbrack {C \subseteq G}\right\rbrack = {p}^{k}. \] (1) How many such cycles \( C = {v}_{0}\ldots {v}_{k - 1}{v}_{0} \) are there? There are \( {\left( n\right) }_{k} \) sequences \( {v}_{0}\ldots {v}_{k - 1} \) of distinct vertices in \( V \), and each cycle is identified by \( {2k} \) of those sequences - so there are exactly \( {\left( n\right) }_{k}/{2k} \) such cycles. Our random variable \( X \) assigns to every graph \( G \) its number of \( k \) - cycles. Clearly, this is the sum of all the values \( {X}_{C}\left( G\right) \), where \( C \) varies over the \( {\left( n\right) }_{k}/{2k} \) cycles of length \( k \) with vertices in \( V \) : \[ X = \mathop{\sum }\limits_{C}{X}_{C} \] Since the expectation is linear, (1) thus implies \[ E\left( X\right) = E\left( {\mathop{\sum }\limits_{C}{X}_{C}}\right) = \mathop{\sum }\limits_{C}E\left( {X}_{C}\right) = \frac{{\left( n\right) }_{k}}{2k}{p}^{k} \] as claimed. ## 11.2 The probabilistic method Very roughly, the probabilistic method in discrete mathematics has developed from the following idea. In order to prove the existence of an object with some desired property, one defines a probability space on some larger - and certainly non-empty - class of objects, and then shows that an element of this space has the desired property with positive probability. The 'objects' inhabiting this probability space may be of any kind: partitions or orderings of the vertices of some fixed graph arise as naturally as mappings, embeddings and, of course, graphs themselves. In this section, we illustrate the probabilistic method by giving a detailed account of one of its earliest results: of Erdős's classic theorem on large girth and chromatic number (Theorem 5.2.5). Erdős’s theorem says that, given any positive integer \( k \), there is a graph \( G \) with girth \( g\left( G\right) > k \) and chromatic number \( \chi \left( G\right) > k \) . Let us call cycles of length at most \( k \) short, and sets of \( \left| G\right| /k \) or more vertices --- short big/small --- big. For a proof of Erdős’s theorem, it suffices to find a graph \( G \) without short cycles and without big independent sets of vertices: then the colour classes in any vertex colouring of \( G \) are small (not big), so we need more than \( k \) colours to colour \( G \) . How can we find such a graph \( G \) ? If we choose \( p \) small enough, then a random graph in \( \mathcal{G}\left( {n, p}\right) \) is unlikely to contain any (short) cycles. If we choose \( p \) large enough, then \( G \) is unlikely to have big independent vertex sets. So the question is: do these two ranges of \( p \) overlap, that is, can we choose \( p \) so that, for some \( n \), it is both small enough to give \( P\left\lbrack {g \leq k}\right\rbrack < \frac{1}{2} \) and large enough for \( P\left\lbrack {\alpha \geq n/k}\right\rbrack < \frac{1}{2} \) ? If so, then \( \mathcal{G}\left( {n, p}\right) \) will contain at least one graph without either short cycles or big independent sets. Unfortunately, such a choice of \( p \) is impossible: the two ranges of \( p \) do not overlap! As we shall see in Section 11.4, we must keep \( p \) below \( {n}^{-1} \) to make the occurrence of short cycles in \( G \) unlikely-but for any such \( p \) there will most likely be no cycles in \( G \) at all (Exercise 18), so \( G \) will be bipartite and hence have at least \( n/2 \) independent vertices. But all is not lost. In order to make big independent sets unlikely, we shall fix \( p \) above \( {n}^{-1} \), at \( {n}^{\epsilon - 1} \) for some \( \epsilon > 0 \) . Fortunately, though, if \( \epsilon \) is small enough then this will produce only few short cycles in \( G \) , even compared with \( n \) (rather than, more typically, with \( {n}^{k} \) ). If we then delete a vertex in each of those cycles, the graph \( H \) obtained will have no short cycles, and its independence number \( \alpha \left( H\right) \) will be at most that of \( G \) . Since \( H \) is not much smaller than \( G \), its chromatic number will thus still be large, so we have found a graph with both large girth and large chromatic number. To prepare for the formal proof of Erdős's theorem, we first show that an edge probability of \( p = {n}^{\epsilon - 1} \) is indeed always large enough to ensure that \( G \in \mathcal{G}\left( {n, p}\right) \) ’almost surely’ has no big independent set of vertices. More precisely, we prove the following slightly stronger assertion: Lemma 11.2.1. Let \( k > 0 \) be an integer, and let \( p = p\left( n\right) \) be a function of \( n \) such that \( p \geq \left( {{6k}\ln n}\right) {n}^{-1} \) for \( n \) large. Then \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}P\left\lbrack {\alpha \geq \frac{1}{2}n/k}\right\rbrack = 0. \] \( \left( {11.1.2}\right) \) Proof. For all integers \( n, r \) with \( n \geq r \geq 2 \), and all \( G \in \mathcal{G}\left( {n, p}\right) \), Lemma 11.1.2 implies \[ P\left\lbrack {\alpha \geq r}\right\rbrack \leq \left( \begin{array}{l} n \\ r \end{array}\right) {q}^{\left( \begin{array}{l} r \\ 2 \end{array}\right) } \] \[ \leq {n}^{r}{q}^{\left( \begin{array}{l} r \\ 2 \end{array}\right) } \] \[ = {\left( n{q}^{\left( {r - 1}\right) /2}\right) }^{r} \] \[ \leq {\left( n{e}^{-p\left( {r - 1}\right) /2}\right) }^{r} \] here, the last inequality follows from the fact that \( 1 - p \leq {e}^{-p} \) for all \( p \) . (Compare the functions \( x \mapsto {e}^{x} \) and \( x \mapsto x + 1 \) for \( x = - p \) .) Now if \( p \geq \left( {{6k}\ln n}\right) {n}^{-1} \) and \( r \geq \frac{1}{2}n/k \), then the term under the exponent satisfies \[ n{e}^{-p\left( {r - 1}\right) /2} = n{e}^{-{pr}/2 + p/2} \] \[ \leq n{e}^{-\left( {3/2}\right) \ln n + p/2} \] \[ \leq n{n}^{-3/2}{e}^{1/2} \] \[ = \sqrt{e}/\sqrt{n}\underset{n \rightarrow \infty }{ \rightarrow }0 \] Since \( p \geq \left( {{6k}\ln n}\right) {n}^{-1} \) for \( n \) large, we thus obtain for \( r \mathrel{\text{:=}} \left\lceil {\frac{1}{2}n/k}\right\rceil \) \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}P\left\lbrack {\alpha \geq \frac{1}{2}n/k}\right\rbrack = \mathop{\lim }\limits_{{n \rightarrow \infty }}P\left\lbrack {\alpha \geq r}\right\rbrack = 0, \] as claimed. We are now ready to prove Theorem 5.2.5, which we restate: Theorem 11.2.2. (Erdős 1959) \( \left\lbrack {9.2.3}\right\rbrack \) For every integer \( k \) there exists a graph \( H \) with girth \( g\left( H\right) > k \) and chromatic number \( \chi \left( H\right) > k \) . (11.1.4) Proof. Assume that \( k \geq 3 \), fix \( \epsilon \) with \( 0 < \epsilon < 1/k \), and let \( p \mathrel{\text{:=}} {n}^{\epsilon - 1} \) . Let (11.1.5) \( X\left( G\right) \) denote the number of short cycles in a random graph \( G \in \mathcal{G}\left( {n, p}\right) \) , \( p,\epsilon, X \) i.e. its number of cycles of length at most \( k \) . By Lemma 11.1.5, we have \[ E\left( X\right) = \mathop{\sum }\limits_{{i = 3}}^{k}\frac{{\left( n\right) }_{i}}{2i}{p}^{i} \leq \frac{1}{2}\mathop{\sum }\limits_{{i = 3}}^{k}{n}^{i}{p}^{i} \leq \frac{1}{2}\left( {k - 2}\right) {n}^{k}{p}^{k}; \] note that \( {\left( np\right) }^{i} \leq {\left( np\right) }^{k} \), because \( {np} = {n}^{\epsilon } \geq 1 \) . By Lemma 11.1.4, \[ P\left\lbrack {X \geq n/2}\right\rbrack \leq E\left( X\right) /\left( {n/2}\right) \] \[ \leq \left( {k - 2}\right) {n}^{k - 1}{p}^{k} \] \[ = \left( {k - 2}\right) {n}^{k - 1}{n}^{\left( {\epsilon - 1}\right) k} \] \[ = \left( {k - 2}\right) {n}^{{k\epsilon } - 1}\text{.} \] As \( {k\epsilon } - 1 < 0 \) by our choice of \( \epsilon \), this implies that \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}P\left\lbrack {X \geq n/2}\right\rbrack = 0. \] Let \( n \) be large enough that \( P\left\lbrack {X \geq n/2}\right\rbrack < \frac{1}{2} \) and \( P\left\lbrack {\alpha \geq \frac{1}{2}n/k}\right\rbrack < \frac{1}{2} \) ; the latter is possible by our choice of \( p \) and Lemma 11.2.1. Then there is a graph \( G \in \mathcal{G}\left( {n, p}\right) \) with fewer than \( n/2 \) short cycles and \( \alpha \left( G\right) < \) \( \frac{1}{2}n/k \) . From each of those cycles delete a vertex, and let \( H \) be the graph obtained. Then \( \left| H\right| \geq n/2 \) and \( H \) has no short

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

rack {X \geq n/2}\right\rbrack \leq E\left( X\right) /\left( {n/2}\right) \] \[ \leq \left( {k - 2}\right) {n}^{k - 1}{p}^{k} \] \[ = \left( {k - 2}\right) {n}^{k - 1}{n}^{\left( {\epsilon - 1}\right) k} \] \[ = \left( {k - 2}\right) {n}^{{k\epsilon } - 1}\text{.} \] As \( {k\epsilon } - 1 < 0 \) by our choice of \( \epsilon \), this implies that \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}P\left\lbrack {X \geq n/2}\right\rbrack = 0. \] Let \( n \) be large enough that \( P\left\lbrack {X \geq n/2}\right\rbrack < \frac{1}{2} \) and \( P\left\lbrack {\alpha \geq \frac{1}{2}n/k}\right\rbrack < \frac{1}{2} \) ; the latter is possible by our choice of \( p \) and Lemma 11.2.1. Then there is a graph \( G \in \mathcal{G}\left( {n, p}\right) \) with fewer than \( n/2 \) short cycles and \( \alpha \left( G\right) < \) \( \frac{1}{2}n/k \) . From each of those cycles delete a vertex, and let \( H \) be the graph obtained. Then \( \left| H\right| \geq n/2 \) and \( H \) has no short cycles, so \( g\left( H\right) > k \) . By definition of \( G \) , \[ \chi \left( H\right) \geq \frac{\left| H\right| }{\alpha \left( H\right) } \geq \frac{n/2}{\alpha \left( G\right) } > k. \] Corollary 11.2.3. There are graphs with arbitrarily large girth and arbitrarily large values of the invariants \( \kappa ,\varepsilon \) and \( \delta \) . Proof. Apply Corollary 5.2.3 and Theorem 1.4.3. ## 11.3 Properties of almost all graphs Recall that a graph property is a class of graphs that is closed under isomorphism, one that contains with every graph \( G \) also the graphs isomorphic to \( G \) . If \( p = p\left( n\right) \) is a fixed function (possibly constant), and \( \mathcal{P} \) is a graph property, we may ask how the probability \( P\left\lbrack {G \in \mathcal{P}}\right\rbrack \) behaves for --- almost all etc. --- \( G \in \mathcal{G}\left( {n, p}\right) \) as \( n \rightarrow \infty \) . If this probability tends to 1, we say that \( G \in \mathcal{P} \) for almost all (or almost every) \( G \in \mathcal{G}\left( {n, p}\right) \), or that \( G \in \mathcal{P} \) almost surely; if it tends to 0, we say that almost no \( G \in \mathcal{G}\left( {n, p}\right) \) has the property \( \mathcal{P} \) . (For example, in Lemma 11.2.1 we proved that, for a certain \( p \), almost no \( G \in \mathcal{G}\left( {n, p}\right) \) has a set of more than \( \frac{1}{2}n/k \) independent vertices.) To illustrate the new concept let us show that, for constant \( p \), every fixed abstract \( {}^{3} \) graph \( H \) is an induced subgraph of almost all graphs: Proposition 11.3.1. For every constant \( p \in \left( {0,1}\right) \) and every graph \( H \) , almost every \( G \in \mathcal{G}\left( {n, p}\right) \) contains an induced copy of \( H \) . Proof. Let \( H \) be given, and \( k \mathrel{\text{:=}} \left| H\right| \) . If \( n \geq k \) and \( U \subseteq \{ 0,\ldots, n - 1\} \) is a fixed set of \( k \) vertices of \( G \), then \( G\left\lbrack U\right\rbrack \) is isomorphic to \( H \) with a certain probability \( r > 0 \) . This probability \( r \) depends on \( p \), but not on \( n \) (why not?). Now \( G \) contains a collection of \( \lfloor n/k\rfloor \) disjoint such sets \( U \) . The probability that none of the corresponding graphs \( G\left\lbrack U\right\rbrack \) is isomorphic to \( H \) is \( {\left( 1 - r\right) }^{\lfloor n/k\rfloor } \), since these events are independent by the disjointness of the edges sets \( {\left\lbrack U\right\rbrack }^{2} \) . Thus \[ P\left\lbrack {H \nsubseteq G\text{ induced }}\right\rbrack \leq {\left( 1 - r\right) }^{\lfloor n/k\rfloor }\underset{n \rightarrow \infty }{ \rightarrow }0, \] which implies the assertion. The following lemma is a simple device enabling us to deduce that quite a number of natural graph properties (including that of Proposition 11.3.1) are shared by almost all graphs. Given \( i, j \in \mathbb{N} \), let \( {\mathcal{P}}_{i, j} \) denote the property that the graph considered contains, for any disjoint vertex sets \( U, W \) with \( \left| U\right| \leq i \) and \( \left| W\right| \leq j \), a vertex \( v \notin U \cup W \) that is adjacent to all the vertices in \( U \) but to none in \( W \) . Lemma 11.3.2. For every constant \( p \in \left( {0,1}\right) \) and \( i, j \in \mathbb{N} \), almost every graph \( G \in \mathcal{G}\left( {n, p}\right) \) has the property \( {\mathcal{P}}_{i, j} \) . --- 3 The word ’abstract’ is used to indicate that only the isomorphism type of \( H \) is known or relevant, not its actual vertex and edge sets. In our context, it indicates that the word 'subgraph' is used in the usual sense of 'isomorphic to a subgraph'. --- Proof. For fixed \( U, W \) and \( v \in G - \left( {U \cup W}\right) \), the probability that \( v \) is adjacent to all the vertices in \( U \) but to none in \( W \), is \[ {p}^{\left| U\right| }{q}^{\left| W\right| } \geq {p}^{i}{q}^{j} \] Hence, the probability that no suitable \( v \) exists for these \( U \) and \( W \), is \[ {\left( 1 - {p}^{\left| U\right| }{q}^{\left| W\right| }\right) }^{n - \left| U\right| - \left| W\right| } \leq {\left( 1 - {p}^{i}{q}^{j}\right) }^{n - i - j} \] (for \( n \geq i + j \) ), since the corresponding events are independent for different \( v \) . As there are no more than \( {n}^{i + j} \) pairs of such sets \( U, W \) in \( V\left( G\right) \) (encode sets \( U \) of fewer than \( i \) points as non-injective maps \( \{ 0,\ldots, i - 1\} \rightarrow \{ 0,\ldots, n - 1\} \), etc.), the probability that some such pair has no suitable \( v \) is at most \[ {n}^{i + j}{\left( 1 - {p}^{i}{q}^{j}\right) }^{n - i - j} \] which tends to zero as \( n \rightarrow \infty \) since \( 1 - {p}^{i}{q}^{j} < 1 \) . Corollary 11.3.3. For every constant \( p \in \left( {0,1}\right) \) and \( k \in \mathbb{N} \), almost every graph in \( \mathcal{G}\left( {n, p}\right) \) is \( k \) -connected. Proof. By Lemma 11.3.2, it is enough to show that every graph in \( {\mathcal{P}}_{2, k - 1} \) is \( k \) -connected. But this is easy: any graph in \( {\mathcal{P}}_{2, k - 1} \) has order at least \( k + 2 \), and if \( W \) is a set of fewer than \( k \) vertices, then by definition of \( {\mathcal{P}}_{2, k - 1} \) any other two vertices \( x, y \) have a common neighbour \( v \notin W \) ; in particular, \( W \) does not separate \( x \) from \( y \) . In the proof of Corollary 11.3.3, we showed substantially more than was asked for: rather than finding, for any two vertices \( x, y \notin W \), some \( x - y \) path avoiding \( W \), we showed that \( x \) and \( y \) have a common neighbour outside \( W \) ; thus, all the paths needed to establish the desired connectivity could in fact be chosen of length 2. What seemed like a clever trick in this particular proof is in fact indicative of a more fundamental phenomenon for constant edge probabilities: by an easy result in logic, any statement about graphs expressed by quantifying over vertices only (rather than over sets or sequences of vertices) \( {}^{4} \) is either almost surely true or almost surely false. All such statements, or their negations, are in fact immediate consequences of an assertion that the graph has property \( {\mathcal{P}}_{i, j} \), for some suitable \( i, j \) . As a last example of an 'almost all' result we now show that almost every graph has a surprisingly high chromatic number: 4 In the terminology of logic: any first order sentence in the language of graph theory Proposition 11.3.4. For every constant \( p \in \left( {0,1}\right) \) and every \( \epsilon > 0 \) , almost every graph \( G \in \mathcal{G}\left( {n, p}\right) \) has chromatic number \[ \chi \left( G\right) > \frac{\log \left( {1/q}\right) }{2 + \epsilon } \cdot \frac{n}{\log n}. \] \( \left( {11.1.2}\right) \) Proof. For any fixed \( n \geq k \geq 2 \), Lemma 11.1.2 implies \[ P\left\lbrack {\alpha \geq k}\right\rbrack \leq \left( \begin{array}{l} n \\ k \end{array}\right) {q}^{\left( \begin{array}{l} k \\ 2 \end{array}\right) } \] \[ \leq {n}^{k}{q}^{\left( \begin{matrix} k \\ 2 \end{matrix}\right) } \] \[ = {q}^{k\frac{\log n}{\log q} + \frac{1}{2}k\left( {k - 1}\right) } \] \[ = {q}^{\frac{k}{2}\left( {-\frac{2\log n}{\log \left( {1/q}\right) } + k - 1}\right) }. \] For \[ k \mathrel{\text{:=}} \left( {2 + \epsilon }\right) \frac{\log n}{\log \left( {1/q}\right) } \] the exponent of this expression tends to infinity with \( n \), so the expression itself tends to zero. Hence, almost every \( G \in \mathcal{G}\left( {n, p}\right) \) is such that in any vertex colouring of \( G \) no \( k \) vertices can have the same colour, so every colouring uses more than \[ \frac{n}{k} = \frac{\log \left( {1/q}\right) }{2 + \epsilon } \cdot \frac{n}{\log n} \] colours. By a result of Bollobás (1988), Proposition 11.3.4 is sharp in the following sense: if we replace \( \epsilon \) by \( - \epsilon \), then the lower bound given for \( \chi \) turns into an upper bound. Most of the results of this section have the interesting common feature that the values of \( p \) played no role whatsoever: if almost every graph in \( \mathcal{G}\left( {n,\frac{1}{2}}\right) \) had the property considered, then the same was true for almost every graph in \( \mathcal{G}\left( {n,1/{1000}}\right) \) . How could this happen? Such insensitivity of our random model to changes of \( p \) was certainly not intended: after all, among all the graphs with a certain property \( \mathcal{P} \) it is often those having \( \mathcal{P} \) ’only just’ that are the most interesting -for those graphs are most likely to have different properties too, properties to which \( \mathcal{P} \) might thus be set in relation. (The proof of Erdős’s theorem is a good example.) For most properties, however-and this explains the above phenomenon—the critical order of magnitude of \( p \) around which the property will 'just' occur or not occur

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d given for \( \chi \) turns into an upper bound. Most of the results of this section have the interesting common feature that the values of \( p \) played no role whatsoever: if almost every graph in \( \mathcal{G}\left( {n,\frac{1}{2}}\right) \) had the property considered, then the same was true for almost every graph in \( \mathcal{G}\left( {n,1/{1000}}\right) \) . How could this happen? Such insensitivity of our random model to changes of \( p \) was certainly not intended: after all, among all the graphs with a certain property \( \mathcal{P} \) it is often those having \( \mathcal{P} \) ’only just’ that are the most interesting -for those graphs are most likely to have different properties too, properties to which \( \mathcal{P} \) might thus be set in relation. (The proof of Erdős’s theorem is a good example.) For most properties, however-and this explains the above phenomenon—the critical order of magnitude of \( p \) around which the property will 'just' occur or not occur lies far below any constant value of \( p \) : it is typically a function of \( n \) tending to zero as \( n \rightarrow \infty \) . Let us then see what happens if \( p \) is allowed to vary with \( n \) . Almost immediately, a fascinating picture unfolds. For edge probabilities \( p \) whose order of magnitude lies below \( {n}^{-2} \), a random graph \( G \in \mathcal{G}\left( {n, p}\right) \) almost surely has no edges at all. As \( p \) grows, \( G \) acquires more and more structure: from about \( p = \sqrt{n}{n}^{-2} \) onwards, it almost surely has a component with more than two vertices, these components grow into trees, and around \( p = {n}^{-1} \) the first cycles are born. Soon, some of these will have several crossing chords, making the graph non-planar. At the same time, one component outgrows the others, until it devours them around \( p = \left( {\log n}\right) {n}^{-1} \), making the graph connected. Hardly later, at \( p = \left( {1 + \epsilon }\right) \left( {\log n}\right) {n}^{-1} \), our graph almost surely has a Hamilton cycle! It has become customary to compare this development of random graphs as \( p \) grows to the evolution of an organism: for each \( p = p\left( n\right) \) , one thinks of the properties shared by almost all graphs in \( \mathcal{G}\left( {n, p}\right) \) as properties of ’the’ typical random graph \( G \in \mathcal{G}\left( {n, p}\right) \), and studies how \( G \) changes its features with the growth rate of \( p \) . As with other species, the evolution of random graphs happens in relatively sudden jumps: the critical edge probabilities mentioned above are thresholds below which almost no graph and above which almost every graph has the property considered. More precisely, we call a real function \( t = t\left( n\right) \) with \( t\left( n\right) \neq 0 \) --- threshold function --- for all \( n \) a threshold function for a graph property \( \mathcal{P} \) if the following holds for all \( p = p\left( n\right) \), and \( G \in \mathcal{G}\left( {n, p}\right) \) : \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}P\left\lbrack {G \in \mathcal{P}}\right\rbrack = \left\{ \begin{array}{ll} 0 & \text{ if }p/t \rightarrow 0\text{ as }n \rightarrow \infty \\ 1 & \text{ if }p/t \rightarrow \infty \text{ as }n \rightarrow \infty . \end{array}\right. \] If \( \mathcal{P} \) has a threshold function \( t \), then clearly any positive multiple \( {ct} \) of \( t \) is also a threshold function for \( \mathcal{P} \) ; thus, threshold functions in the above sense are only ever unique up to a multiplicative constant. \( {}^{5} \) Which graph properties have threshold functions? Natural candidates for such properties are increasing ones, properties closed under the addition of edges. (Graph properties of the form \( \{ G \mid G \supseteq H\} \), with \( H \) fixed, are common increasing properties; connectedness is another.) And indeed, Bollobás & Thomason (1987) have shown that all increasing properties, trivial exceptions aside, have threshold functions. In the next section we shall study a general method to compute threshold functions. We finish this section with a little gem, the one and only theorem about infinite random graphs. Let \( \mathcal{G}\left( {{\aleph }_{0}, p}\right) \) be defined exactly like \( \mathcal{G}\left( {n, p}\right) \) for \( n = {\aleph }_{0} \), as the (product) space of random graphs on \( \mathbb{N} \) whose edges are chosen independently with probability \( p \) . --- 5 Our notion of threshold reflects only the crudest interesting level of screening: for some properties, such as connectedness, one can define sharper thresholds where the constant factor is crucial. Note also the role of the constant factor in our comparison of connectedness with hamiltonicity in the previous paragraph. --- As we saw in Lemma 11.3.2, the properties \( {\mathcal{P}}_{i, j} \) hold almost surely for finite random graphs with constant edge probability. It will therefore hardly come as a surprise that an infinite random graph almost surely (which now has the usual meaning of 'with probability 1') has all these properties at once. However, in Chapter 8.3 we saw that, up to isomorphism, there is exactly one countable graph, the Rado graph \( R \), that has property \( {\mathcal{P}}_{i, j} \) for all \( i, j \in \mathbb{N} \) simultaneously; this joint property was denoted as \( \left( *\right) \) there. Combining these facts, we get the following rather bizarre result: Theorem 11.3.5. (Erdős and Rényi 1963) With probability 1, a random graph \( G \in \mathcal{G}\left( {{\aleph }_{0}, p}\right) \) with \( 0 < p < 1 \) is isomorphic to the Rado graph \( R \) . (8.3.1) Proof. Given fixed disjoint finite sets \( U, W \subseteq \mathbb{N} \), the probability that a vertex \( v \notin U \cup W \) is not joined to \( U \cup W \) as expressed in property \( \left( *\right) \) of Chapter 8.3 (i.e., is not joined to all of \( U \) or is joined to some vertex in \( W \) ) is some number \( r < 1 \) depending only on \( U \) and \( W \) . The probability that none of \( k \) given vertices \( v \) is joined to \( U \cup W \) as in \( \left( *\right) \) is \( {r}^{k} \), which tends to 0 as \( k \rightarrow \infty \) . Hence the probability that all the (infinitely many) vertices outside \( U \cup W \) fail to witness \( \left( *\right) \) for these sets \( U \) and \( W \) is 0 . Now there are only countably many choices for \( U \) and \( W \) as above. Since the union of countably many sets of measure 0 again has measure 0 , the probability that \( \left( *\right) \) fails for any sets \( U \) and \( W \) is still 0 . Therefore \( G \) satisfies \( \left( *\right) \) with probability 1. By Theorem 8.3.1 this means that, almost surely, \( G \simeq R \) . How can we make sense of the paradox that the result of infinitely many independent choices can be so predictable? The answer, of course, lies in the fact that the uniqueness of \( R \) holds only up to isomorphism. Now, constructing an automorphism for an infinite graph with property \( \left( *\right) \) is a much easier task than finding one for a finite random graph, so in this sense the uniqueness is no longer that surprising. Viewed in this way, Theorem 11.3.5 expresses not a lack of variety in infinite random graphs but rather the abundance of symmetry that glosses over this variety when the graphs \( G \in \mathcal{G}\left( {{\aleph }_{0}, p}\right) \) are viewed only up to isomorphism. ## 11.4 Threshold functions and second moments Consider a graph property of the form \[ \mathcal{P} = \{ G \mid X\left( G\right) \geq 1\} \] where \( X \geq 0 \) is a random variable on \( \mathcal{G}\left( {n, p}\right) \) . Many properties can be expressed naturally in this way; if \( X \) denotes the number of spanning trees, for example, then \( \mathcal{P} \) corresponds to connectedness. How could we prove that \( \mathcal{P} \) has a threshold function \( t \) ? Any such proof will consist of two parts: a proof that almost no \( G \in \mathcal{G}\left( {n, p}\right) \) has \( \mathcal{P} \) when \( p \) is small compared with \( t \), and one showing that almost every \( G \) has \( \mathcal{P} \) when \( p \) is large. Since \( X \geq 0 \), we may use Markov’s inequality for the first part of the proof and find an upper bound for \( E\left( X\right) \) instead of \( P\left\lbrack {X \geq 1}\right\rbrack \) : if \( E\left( X\right) \) is much smaller than 1 then \( X\left( G\right) \) can be at least 1 only for few \( G \in \mathcal{G}\left( {n, p}\right) \) , and for almost no \( G \) if \( E\left( X\right) \rightarrow 0 \) as \( n \rightarrow \infty \) . Besides, the expectation is much easier to calculate than probabilities: without worrying about such things as independence or incompatibility of events, we may compute the expectation of a sum of random variables - for example, of indicator random variables - simply by adding up their individual expected values. For the second part of the proof, things are more complicated. In order to show that \( P\left\lbrack {X \geq 1}\right\rbrack \) is large, it is not enough to bound \( E\left( X\right) \) from below: since \( X \) is not bounded above, \( E\left( X\right) \) may be large simply because \( X \) is very large on just a few graphs \( G \) - so \( X \) may still be zero for most \( G \in \mathcal{G}\left( {n, p}\right) .{}^{6} \) In order to prove that \( P\left\lbrack {X \geq 1}\right\rbrack \rightarrow 1 \), we thus have to show that this cannot happen, i.e. that \( X \) does not deviate a lot from its mean too often. The following elementary tool from probability theory achieves just that. As is customary, we write \[ \mu \mathrel{\text{:=}} E\left( X\right) \] and define \( \sigma \geq 0 \) by setting \[ {\sigma }^{2} \mathrel{\text{:=}} E\left( {\left( X - \mu \right) }^{2}\right) \] This quantity \( {\sigma }^{2} \) is called the variance or second m

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individual expected values. For the second part of the proof, things are more complicated. In order to show that \( P\left\lbrack {X \geq 1}\right\rbrack \) is large, it is not enough to bound \( E\left( X\right) \) from below: since \( X \) is not bounded above, \( E\left( X\right) \) may be large simply because \( X \) is very large on just a few graphs \( G \) - so \( X \) may still be zero for most \( G \in \mathcal{G}\left( {n, p}\right) .{}^{6} \) In order to prove that \( P\left\lbrack {X \geq 1}\right\rbrack \rightarrow 1 \), we thus have to show that this cannot happen, i.e. that \( X \) does not deviate a lot from its mean too often. The following elementary tool from probability theory achieves just that. As is customary, we write \[ \mu \mathrel{\text{:=}} E\left( X\right) \] and define \( \sigma \geq 0 \) by setting \[ {\sigma }^{2} \mathrel{\text{:=}} E\left( {\left( X - \mu \right) }^{2}\right) \] This quantity \( {\sigma }^{2} \) is called the variance or second moment of \( X \) ; by definition, it is a (quadratic) measure of how much \( X \) deviates from its mean. Since \( E \) is linear, the defining term for \( {\sigma }^{2} \) expands to \[ {\sigma }^{2} = E\left( {{X}^{2} - {2\mu X} + {\mu }^{2}}\right) = E\left( {X}^{2}\right) - {\mu }^{2}. \] Note that \( \mu \) and \( {\sigma }^{2} \) always refer to a random variable on some fixed probability space. In our setting, where we consider the spaces \( \mathcal{G}\left( {n, p}\right) \) , both quantities are functions of \( n \) . The following lemma says exactly what we need: that \( X \) cannot deviate a lot from its mean too often. 6 For some \( p \) between \( {n}^{-1} \) and \( \left( {\log n}\right) {n}^{-1} \), for example, almost every \( G \in \mathcal{G}\left( {n, p}\right) \) has an isolated vertex (and hence no spanning tree), but its expected number of spanning trees tends to infinity with \( n \) . See the Exercise 12 for details. Lemma 11.4.1. (Chebyshev's Inequality) For all real \( \lambda > 0 \) , \[ P\left\lbrack {\left| {X - \mu }\right| \geq \lambda }\right\rbrack \leq {\sigma }^{2}/{\lambda }^{2} \] \( \left( {11.1.4}\right) \) Proof. By Lemma 11.1.4 and definition of \( {\sigma }^{2} \) , \[ P\left\lbrack {\left| {X - \mu }\right| \geq \lambda }\right\rbrack = P\left\lbrack {{\left( X - \mu \right) }^{2} \geq {\lambda }^{2}}\right\rbrack \leq {\sigma }^{2}/{\lambda }^{2}. \] For a proof that \( X\left( G\right) \geq 1 \) for almost all \( G \in \mathcal{G}\left( {n, p}\right) \), Chebyshev’s inequality can be used as follows: Lemma 11.4.2. If \( \mu > 0 \) for \( n \) large, and \( {\sigma }^{2}/{\mu }^{2} \rightarrow 0 \) as \( n \rightarrow \infty \), then \( X\left( G\right) > 0 \) for almost all \( G \in \mathcal{G}\left( {n, p}\right) \) . Proof. Any graph \( G \) with \( X\left( G\right) = 0 \) satisfies \( \left| {X\left( G\right) - \mu }\right| = \mu \) . Hence Lemma 11.4.1 implies with \( \lambda \mathrel{\text{:=}} \mu \) that \[ P\left\lbrack {X = 0}\right\rbrack \leq P\left\lbrack {\left| {X - \mu }\right| \geq \mu }\right\rbrack \leq {\sigma }^{2}/{\mu }^{2}\underset{n \rightarrow \infty }{ \rightarrow }0. \] Since \( X \geq 0 \), this means that \( X > 0 \) almost surely, i.e. that \( X\left( G\right) > 0 \) for almost all \( G \in \mathcal{G}\left( {n, p}\right) \) . As the main result of this section, we now prove a theorem that will at once give us threshold functions for a number of natural properties. \( {\mathcal{P}}_{H} \) Given a graph \( H \), we denote by \( {\mathcal{P}}_{H} \) the graph property of containing a balanced copy of \( H \) as a subgraph. We shall call \( H \) balanced if \( \varepsilon \left( {H}^{\prime }\right) \leq \varepsilon \left( H\right) \) for all subgraphs \( {H}^{\prime } \) of \( H \) . Theorem 11.4.3. (Erdős & Rényi 1960) \( k,\ell \) If \( H \) is a balanced graph with \( k \) vertices and \( \ell \geq 1 \) edges, then \( t\left( n\right) \mathrel{\text{:=}} \) \( t \) \( {n}^{-k/\ell } \) is a threshold function for \( {\mathcal{P}}_{H} \) . (11.1.4) Proof. Let \( X\left( G\right) \) denote the number of subgraphs of \( G \) isomorphic to \( H \) . (11.1.5) Given \( n \in \mathbb{N} \), let \( \mathcal{H} \) denote the set of all graphs isomorphic to \( H \) whose \( X \) vertices lie in \( \{ 0,\ldots, n - 1\} \), the vertex set of the graphs \( G \in \mathcal{G}\left( {n, p}\right) \) : \( \mathcal{H} \) \[ \mathcal{H} \mathrel{\text{:=}} \left\{ {{H}^{\prime } \mid {H}^{\prime } \simeq H, V\left( {H}^{\prime }\right) \subseteq \{ 0,\ldots, n - 1\} }\right\} . \] Given \( {H}^{\prime } \in \mathcal{H} \) and \( G \in \mathcal{G}\left( {n, p}\right) \), we shall write \( {H}^{\prime } \subseteq G \) to express that \( {H}^{\prime } \) itself - not just an isomorphic copy of \( {H}^{\prime } \) -is a subgraph of \( G \) . \( h \) By \( h \) we denote the number of isomorphic copies of \( H \) on a fixed \( k \) -set; clearly, \( h \leq k \) !. As there are \( \left( \begin{array}{l} n \\ k \end{array}\right) \) possible vertex sets for the graphs in \( \mathcal{H} \), we thus have \[ \left| \mathcal{H}\right| = \left( \begin{array}{l} n \\ k \end{array}\right) h \leq \left( \begin{array}{l} n \\ k \end{array}\right) k! \leq {n}^{k} \] (1) Given \( p = p\left( n\right) \), we set \( \gamma \mathrel{\text{:=}} p/t \) ; then \[ p = \gamma {n}^{-k/\ell }. \] (2) We have to show that almost no \( G \in \mathcal{G}\left( {n, p}\right) \) lies in \( {\mathcal{P}}_{H} \) if \( \gamma \rightarrow 0 \) as \( n \rightarrow \infty \) , and that almost all \( G \in \mathcal{G}\left( {n, p}\right) \) lie in \( {\mathcal{P}}_{H} \) if \( \gamma \rightarrow \infty \) as \( n \rightarrow \infty \) . For the first part of the proof, we find an upper bound for \( E\left( X\right) \), the expected number of subgraphs of \( G \) isomorphic to \( H \) . As in the proof of Lemma 11.1.5, double counting gives \[ E\left( X\right) = \mathop{\sum }\limits_{{{H}^{\prime } \in \mathcal{H}}}P\left\lbrack {{H}^{\prime } \subseteq G}\right\rbrack \] (3) For every fixed \( {H}^{\prime } \in \mathcal{H} \), we have \[ P\left\lbrack {{H}^{\prime } \subseteq G}\right\rbrack = {p}^{\ell }, \] (4) because \( \parallel H\parallel = \ell \) . Hence, \[ E\left( X\right) \underset{\left( 3,4\right) }{ = }\left| \mathcal{H}\right| {p}^{\ell }\underset{\left( 1,2\right) }{ \leq }{n}^{k}{\left( \gamma {n}^{-k/\ell }\right) }^{\ell } = {\gamma }^{\ell }. \] (5) Thus if \( \gamma \rightarrow 0 \) as \( n \rightarrow \infty \), then \[ P\left\lbrack {G \in {\mathcal{P}}_{H}}\right\rbrack = P\left\lbrack {X \geq 1}\right\rbrack \leq E\left( X\right) \leq {\gamma }^{\ell }\underset{n \rightarrow \infty }{ \rightarrow }0 \] by Markov’s inequality (11.1.4), so almost no \( G \in \mathcal{G}\left( {n, p}\right) \) lies in \( {\mathcal{P}}_{H} \) . We now come to the second part of the proof: we show that almost all \( G \in \mathcal{G}\left( {n, p}\right) \) lie in \( {\mathcal{P}}_{H} \) if \( \gamma \rightarrow \infty \) as \( n \rightarrow \infty \) . Note first that, for \( n \geq k \) , \[ \left( \begin{array}{l} n \\ k \end{array}\right) {n}^{-k} = \frac{1}{k!}\left( {\frac{n}{n}\cdots \frac{n - k + 1}{n}}\right) \] \[ \geq \frac{1}{k!}{\left( \frac{n - k + 1}{n}\right) }^{k} \] \[ \geq \frac{1}{k!}{\left( 1 - \frac{k - 1}{k}\right) }^{k} \] (6) thus, \( {n}^{k} \) exceeds \( \left( \begin{array}{l} n \\ k \end{array}\right) \) by no more than a factor independent of \( n \) . Our goal is to apply Lemma 11.4.2, and hence to bound \( {\sigma }^{2}/{\mu }^{2} = \) \( \left( {E\left( {X}^{2}\right) - {\mu }^{2}}\right) /{\mu }^{2} \) from above. As in (3) we have \[ E\left( {X}^{2}\right) = \mathop{\sum }\limits_{{\left( {{H}^{\prime },{H}^{\prime \prime }}\right) \in {\mathcal{H}}^{2}}}P\left\lbrack {{H}^{\prime } \cup {H}^{\prime \prime } \subseteq G}\right\rbrack . \] (7) Let us then calculate these probabilities \( P\left\lbrack {{H}^{\prime } \cup {H}^{\prime \prime } \subseteq G}\right\rbrack \) . Given \( {H}^{\prime },{H}^{\prime \prime } \in \mathcal{H} \), we have \[ P\left\lbrack {{H}^{\prime } \cup {H}^{\prime \prime } \subseteq G}\right\rbrack = {p}^{2\ell - \begin{Vmatrix}{{H}^{\prime } \cap {H}^{\prime \prime }}\end{Vmatrix}}. \] Since \( H \) is balanced, \( \varepsilon \left( {{H}^{\prime } \cap {H}^{\prime \prime }}\right) \leq \varepsilon \left( H\right) = \ell /k \) . With \( \left| {{H}^{\prime } \cap {H}^{\prime \prime }}\right| = : i \) this yields \( \begin{Vmatrix}{{H}^{\prime } \cap {H}^{\prime \prime }}\end{Vmatrix} \leq i\ell /k \), so by \( 0 \leq p \leq 1 \) , \[ P\left\lbrack {{H}^{\prime } \cup {H}^{\prime \prime } \subseteq G}\right\rbrack \leq {p}^{2\ell - i\ell /k}. \] (8) We have now estimated the individual summands in (7); what does this imply for the sum as a whole? Since (8) depends on the parameter \( i = \left| {{H}^{\prime } \cap {H}^{\prime \prime }}\right| \), we partition the range \( {\mathcal{H}}^{2} \) of the sum in (7) into the subsets \( {\mathcal{H}}_{i}^{2} \) \[ {\mathcal{H}}_{i}^{2} \mathrel{\text{:=}} \left\{ {\left( {{H}^{\prime },{H}^{\prime \prime }}\right) \in {\mathcal{H}}^{2} : \left| {{H}^{\prime } \cap {H}^{\prime \prime }}\right| = i}\right\} ,\;i = 0,\ldots, k, \] and calculate for each \( {\mathcal{H}}_{i}^{2} \) the corresponding sum \( {A}_{i} \) \[ {A}_{i} \mathrel{\text{:=}} \mathop{\sum }\limits_{i}P\left\lbrack {{H}^{\prime } \cup {H}^{\prime \prime } \subseteq G}\right\rbrack \] \( \mathop{\sum }\limits_{i} \) by itself. (Here, as below, we use \( \mathop{\sum }\limits_{i} \) to denote sums over all pairs \( \left( {{H}^{\prime },{H}^{\prime \prime }}\right) \in {\mathcal{H}}_{i}^{2} \) .) If \( i = 0 \) then \( {H}^{\prime } \) and \( {H}^{\prime \prime } \) are disjoint, so the events \( {H}^{\prime } \subseteq G \) and \( {H}^{\prime \prime } \subseteq G \) are indepe

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\cap {H}^{\prime \prime }}\right| \), we partition the range \( {\mathcal{H}}^{2} \) of the sum in (7) into the subsets \( {\mathcal{H}}_{i}^{2} \) \[ {\mathcal{H}}_{i}^{2} \mathrel{\text{:=}} \left\{ {\left( {{H}^{\prime },{H}^{\prime \prime }}\right) \in {\mathcal{H}}^{2} : \left| {{H}^{\prime } \cap {H}^{\prime \prime }}\right| = i}\right\} ,\;i = 0,\ldots, k, \] and calculate for each \( {\mathcal{H}}_{i}^{2} \) the corresponding sum \( {A}_{i} \) \[ {A}_{i} \mathrel{\text{:=}} \mathop{\sum }\limits_{i}P\left\lbrack {{H}^{\prime } \cup {H}^{\prime \prime } \subseteq G}\right\rbrack \] \( \mathop{\sum }\limits_{i} \) by itself. (Here, as below, we use \( \mathop{\sum }\limits_{i} \) to denote sums over all pairs \( \left( {{H}^{\prime },{H}^{\prime \prime }}\right) \in {\mathcal{H}}_{i}^{2} \) .) If \( i = 0 \) then \( {H}^{\prime } \) and \( {H}^{\prime \prime } \) are disjoint, so the events \( {H}^{\prime } \subseteq G \) and \( {H}^{\prime \prime } \subseteq G \) are independent. Hence, \[ {A}_{0} = \mathop{\sum }\limits_{0}P\left\lbrack {{H}^{\prime } \cup {H}^{\prime \prime } \subseteq G}\right\rbrack \] \[ = \mathop{\sum }\limits_{0}P\left\lbrack {{H}^{\prime } \subseteq G}\right\rbrack \cdot P\left\lbrack {{H}^{\prime \prime } \subseteq G}\right\rbrack \] \[ \leq \mathop{\sum }\limits_{{\left( {{H}^{\prime },{H}^{\prime \prime }}\right) \in {\mathcal{H}}^{2}}}P\left\lbrack {{H}^{\prime } \subseteq G}\right\rbrack \cdot P\left\lbrack {{H}^{\prime \prime } \subseteq G}\right\rbrack \] \[ = \left( {\mathop{\sum }\limits_{{{H}^{\prime } \in \mathcal{H}}}P\left\lbrack {{H}^{\prime } \subseteq G}\right\rbrack }\right) \cdot \left( {\mathop{\sum }\limits_{{{H}^{\prime \prime } \in \mathcal{H}}}P\left\lbrack {{H}^{\prime \prime } \subseteq G}\right\rbrack }\right) \] \[ \underset{\left( 3\right) }{ = }{\mu }^{2}\text{. } \] (9) Let us now estimate \( {A}_{i} \) for \( i \geq 1 \) . Writing \( \mathop{\sum }\limits^{\prime } \) for \( \mathop{\sum }\limits_{{{H}^{\prime } \in \mathcal{H}}} \) and \( \mathop{\sum }\limits^{{\prime \prime }} \) \( \mathop{\sum }\limits^{\prime } \) for \( \mathop{\sum }\limits_{{{H}^{\prime \prime } \in \mathcal{H}}} \), we note that \( \mathop{\sum }\limits_{i} \) can be written as \( \mathop{\sum }\limits^{\prime }\mathop{\sum }\limits_{{\left| {{H}^{\prime } \cap {H}^{\prime \prime }}\right| = i}}^{{\prime \prime }} \) . For fixed \( {H}^{\prime } \) (corresponding to the first sum \( \mathop{\sum }\limits^{\prime } \) ), the second sum ranges over \[ \left( \begin{array}{l} k \\ i \end{array}\right) \left( \begin{matrix} n - k \\ k - i \end{matrix}\right) h \] summands: the number of graphs \( {H}^{\prime \prime } \in \mathcal{H} \) with \( \left| {{H}^{\prime \prime } \cap {H}^{\prime }}\right| = i \) . Hence, for all \( i \geq 1 \) and suitable constants \( {c}_{1},{c}_{2} \) independent of \( n \) , \[ {A}_{i} = \mathop{\sum }\limits_{i}P\left\lbrack {{H}^{\prime } \cup {H}^{\prime \prime } \subseteq G}\right\rbrack \] \[ \underset{\left( 8\right) }{ \leq }\mathop{\sum }\limits^{\prime }\left( \begin{matrix} k \\ i \end{matrix}\right) \left( \begin{matrix} n - k \\ k - i \end{matrix}\right) h{p}^{2\ell }{p}^{-i\ell /k} \] \[ \underset{\left( 2\right) }{ = }\left| \mathcal{H}\right| \left( \begin{matrix} k \\ i \end{matrix}\right) \left( \begin{matrix} n - k \\ k - i \end{matrix}\right) h{p}^{2\ell }{\left( \gamma {n}^{-k/\ell }\right) }^{-i\ell /k} \] \[ \leq \left| \mathcal{H}\right| {p}^{\ell }{c}_{1}{n}^{k - i}h{p}^{\ell }{\gamma }^{-i\ell /k}{n}^{i} \] \[ \underset{\left( 5\right) }{ = }\mu {c}_{1}{n}^{k}h{p}^{\ell }{\gamma }^{-i\ell /k} \] \[ \underset{\left( 6\right) }{ \leq }\mu {c}_{2}\left( \begin{array}{l} n \\ k \end{array}\right) h{p}^{\ell }{\gamma }^{-i\ell /k} \] \[ \underset{\left( 1,5\right) }{ = }{\mu }^{2}{c}_{2}{\gamma }^{-i\ell /k} \] \[ \leq {\mu }^{2}{c}_{2}{\gamma }^{-\ell /k} \] if \( \gamma \geq 1 \) . By definition of the \( {A}_{i} \), this implies with \( {c}_{3} \mathrel{\text{:=}} k{c}_{2} \) that \[ E\left( {X}^{2}\right) /{\mu }^{2}\underset{\left( 7\right) }{ = }\left( {{A}_{0}/{\mu }^{2} + \mathop{\sum }\limits_{{i = 1}}^{k}{A}_{i}/{\mu }^{2}}\right) \underset{\left( 9\right) }{ \leq }1 + {c}_{3}{\gamma }^{-\ell /k} \] and hence \[ \frac{{\sigma }^{2}}{{\mu }^{2}} = \frac{E\left( {X}^{2}\right) - {\mu }^{2}}{{\mu }^{2}} \leq {c}_{3}{\gamma }^{-\ell /k}\underset{\gamma \rightarrow \infty }{ \rightarrow }0. \] By Lemma 11.4.2, therefore, \( X > 0 \) almost surely, i.e. almost all \( G \in \) \( \mathcal{G}\left( {n, p}\right) \) have a subgraph isomorphic to \( H \) and hence lie in \( {\mathcal{P}}_{H} \) . Theorem 11.4.3 allows us to read off threshold functions for a number of natural graph properties. Corollary 11.4.4. If \( k \geq 3 \), then \( t\left( n\right) = {n}^{-1} \) is a threshold function for the property of containing a \( k \) -cycle. Interestingly, the threshold function in Corollary 11.4.4 is independent of the cycle length \( k \) considered: in the evolution of random graphs, cycles of all (constant) lengths appear at about the same time! There is a similar phenomenon for trees. Here, the threshold function does depend on the order of the tree considered, but not on its shape: Corollary 11.4.5. If \( T \) is a tree of order \( k \geq 2 \), then \( t\left( n\right) = {n}^{-k/\left( {k - 1}\right) } \) is a threshold function for the property of containing a copy of \( T \) . We finally have the following result for complete subgraphs: Corollary 11.4.6. If \( k \geq 2 \), then \( t\left( n\right) = {n}^{-2/\left( {k - 1}\right) } \) is a threshold function for the property of containing a \( {K}^{k} \) . Proof. \( {K}^{k} \) is balanced, because \( \varepsilon \left( {K}^{i}\right) = \frac{1}{2}\left( {i - 1}\right) < \frac{1}{2}\left( {k - 1}\right) = \varepsilon \left( {K}^{k}\right) \) for \( i < k \) . With \( \ell \mathrel{\text{:=}} \begin{Vmatrix}{K}^{k}\end{Vmatrix} = \frac{1}{2}k\left( {k - 1}\right) \), we obtain \( {n}^{-k/\ell } = {n}^{-2/\left( {k - 1}\right) } \) . It is not difficult to adapt the proof of Theorem 11.4.3 to the case that \( H \) is unbalanced. The threshold then becomes \( t\left( n\right) = {n}^{-1/{\varepsilon }^{\prime }\left( H\right) } \) , where \( {\varepsilon }^{\prime }\left( H\right) \mathrel{\text{:=}} \max \{ \varepsilon \left( F\right) \mid F \subseteq H\} \) ; see Exercise 21. ## Exercises 1. \( {}^{ - } \) What is the probability that a random graph in \( \mathcal{G}\left( {n, p}\right) \) has exactly \( m \) edges, for \( 0 \leq m \leq \left( \begin{array}{l} n \\ 2 \end{array}\right) \) fixed? 2. What is the expected number of edges in \( G \in \mathcal{G}\left( {n, p}\right) \) ? 3. What is the expected number of \( {K}^{r} \) -subgraphs in \( G \in \mathcal{G}\left( {n, p}\right) \) ? 4. Characterize the graphs that occur as a subgraph in every graph of sufficiently large average degree. 5. In the usual terminology of measure spaces (and in particular, of probability spaces), the phrase 'almost all' is used to refer to a set of points whose complement has measure zero. Rather than considering a limit of probabilities in \( \mathcal{G}\left( {n, p}\right) \) as \( n \rightarrow \infty \), would it not be more natural to define a probability space on the set of all finite graphs (one copy of each) and to investigate properties of 'almost all' graphs in this space, in the sense above? 6. Show that if almost all \( G \in \mathcal{G}\left( {n, p}\right) \) have a graph property \( {\mathcal{P}}_{1} \) and almost all \( G \in \mathcal{G}\left( {n, p}\right) \) have a graph property \( {\mathcal{P}}_{2} \), then almost all \( G \in \mathcal{G}\left( {n, p}\right) \) have both properties, i.e. have the property \( {\mathcal{P}}_{1} \cap {\mathcal{P}}_{2} \) . 7. \( {}^{ - } \) Show that, for constant \( p \in \left( {0,1}\right) \), almost every graph in \( \mathcal{G}\left( {n, p}\right) \) has diameter 2. 8. Show that, for constant \( p \in \left( {0,1}\right) \), almost no graph in \( \mathcal{G}\left( {n, p}\right) \) has a separating complete subgraph. 9. Derive Proposition 11.3.1 from Lemma 11.3.2. 10. Let \( \epsilon > 0 \) and \( p = p\left( n\right) > 0 \), and let \( r \geq \left( {1 + \epsilon }\right) \left( {2\ln n}\right) /p \) be an integer-valued function of \( n \) . Show that almost no graph in \( \mathcal{G}\left( {n, p}\right) \) contains \( r \) independent vertices. 11. Show that for every graph \( H \) there exists a function \( p = p\left( n\right) \) such that \( \mathop{\lim }\limits_{{n \rightarrow \infty }}p\left( n\right) = 0 \) but almost every \( G \in \mathcal{G}\left( {n, p}\right) \) contains an induced copy of \( H \) . 12. \( {}^{ + } \) (i) Show that, for every \( 0 < \epsilon \leq 1 \) and \( p = \left( {1 - \epsilon }\right) \left( {\ln n}\right) {n}^{-1} \), almost every \( G \in \mathcal{G}\left( {n, p}\right) \) has an isolated vertex. (ii) Find a probability \( p = p\left( n\right) \) such that almost every \( G \in \mathcal{G}\left( {n, p}\right) \) is disconnected but the expected number of spanning trees of \( G \) tends to infinity as \( n \rightarrow \infty \) . (Hint for (ii): A theorem of Cayley states that \( {K}^{n} \) has exactly \( {n}^{n - 2} \) spanning trees.) 13. \( {}^{ + } \) Given \( r \in \mathbb{N} \), find a \( c > 0 \) such that, for \( p = c{n}^{-1} \), almost every \( G \in \mathcal{G}\left( {n, p}\right) \) has a \( {K}^{r} \) minor. Can \( c \) be chosen independently of \( r \) ? 14. Find an increasing graph property without a threshold function, and a property that is not increasing but has a threshold function. 15. \( {}^{ - } \) Let \( H \) be a graph of order \( k \), and let \( h \) denote the number of graphs isomorphic to \( H \) on some fixed set of \( k \) elements. Show that \( h \leq k \) !. For which graphs \( H \) does equality hold? 16. \(

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t) \) has an isolated vertex. (ii) Find a probability \( p = p\left( n\right) \) such that almost every \( G \in \mathcal{G}\left( {n, p}\right) \) is disconnected but the expected number of spanning trees of \( G \) tends to infinity as \( n \rightarrow \infty \) . (Hint for (ii): A theorem of Cayley states that \( {K}^{n} \) has exactly \( {n}^{n - 2} \) spanning trees.) 13. \( {}^{ + } \) Given \( r \in \mathbb{N} \), find a \( c > 0 \) such that, for \( p = c{n}^{-1} \), almost every \( G \in \mathcal{G}\left( {n, p}\right) \) has a \( {K}^{r} \) minor. Can \( c \) be chosen independently of \( r \) ? 14. Find an increasing graph property without a threshold function, and a property that is not increasing but has a threshold function. 15. \( {}^{ - } \) Let \( H \) be a graph of order \( k \), and let \( h \) denote the number of graphs isomorphic to \( H \) on some fixed set of \( k \) elements. Show that \( h \leq k \) !. For which graphs \( H \) does equality hold? 16. \( {}^{ - } \) For every \( k \geq 1 \), find a threshold function for \( \{ G \mid \Delta \left( G\right) \geq k\} \) . 17. \( {}^{ - } \) Given \( d \in \mathbb{N} \), is there a threshold function for the property of containing a \( d \) -dimensional cube (see Ex. 2, Ch. 1)? If so, which; if not, why not? 18. Show that \( t\left( n\right) = {n}^{-1} \) is also a threshold function for the property of containing any cycle. 19. Does the property of containing any tree of order \( k \) (for \( k \geq 2 \) fixed) have a threshold function? If so, which? 20. \( {}^{ + } \) Given a graph \( H \), let \( \mathcal{P} \) be the property of containing an induced copy of \( H \) . If \( H \) is complete then, by Corollary 11.4.6, \( \mathcal{P} \) has a threshold function. Show that \( \mathcal{P} \) has no threshold function if \( H \) is not complete. 21. \( {}^{ + } \) Prove the following version of Theorem 11.4.3 for unbalanced subgraphs. Let \( H \) be any graph with at least one edge, and put \( {\varepsilon }^{\prime }\left( H\right) \mathrel{\text{:=}} \) \( \max \{ \varepsilon \left( F\right) \mid \varnothing \neq F \subseteq H\} \) . Then the threshold function for \( {\mathcal{P}}_{H} \) is \( t\left( n\right) = {n}^{-1/{\varepsilon }^{\prime }\left( H\right) } \) . ## Notes There are a number of monographs and texts on the subject of random graphs. The first comprehensive monograph was B. Bollobás, Random Graphs, Academic Press 1985. Another advanced but very readable monograph is S. Janson, T. Luczak & A. Ruciński, Random Graphs, Wiley 2000; this concentrates on areas developed since Random Graphs was published. E.M. Palmer, Graphical Evolution, Wiley 1985, covers material similar to parts of Random Graphs but is written in a more elementary way. Compact introductions going beyond what is covered in this chapter are given by B. Bollobás, Graph Theory, Springer GTM 63, 1979, and by M. Karoński, Handbook of Combinatorics (R.L. Graham, M. Grötschel & L. Lovász, eds.), North-Holland 1995. A stimulating advanced introduction to the use of random techniques in discrete mathematics more generally is given by N. Alon & J.H. Spencer, The Probabilistic Method, Wiley 1992. One of the attractions of this book lies in the way it shows probabilistic methods to be relevant in proofs of entirely deterministic theorems, where nobody would suspect it. Other examples for this phenomenon are Alon's proof of Theorem 5.4.1, or the proof of Theorem 1.3.4; see the notes for Chapters 5 and 1, respectively. The probabilistic method had its first origins in the 1940s, one of its earliest results being Erdős's probabilistic lower bound for Ramsey numbers (Theorem 11.1.3). Lemma 11.3.2 about the properties \( {\mathcal{P}}_{i, j} \) is taken from Bol-lobás's Springer text cited above. A very readable rendering of the proof that, for constant \( p \), every first order sentence about graphs is either almost surely true or almost surely false, is given by P. Winkler, Random structures and zero-one laws, in (N.W. Sauer et al., eds.) Finite and Infinite Combinatorics in Sets and Logic (NATO ASI Series C 411), Kluwer 1993. Theorem 11.3.5 is due to P. Erdős and A. Rényi, Asymmetric graphs, Acta Math. Acad. Sci. Hungar. 14 (1963), 295-315. For further references about the infinite random graph \( R \) see the notes in Chapter 8. The seminal paper on graph evolution is P. Erdős & A. Rényi, On the evolution of random graphs, Publ. Math. Inst. Hungar. Acad. Sci. 5 (1960), 17-61. This paper also includes Theorem 11.4.3 and its proof. The generalization of this theorem to unbalanced subgraphs was first proved by Bollobás in 1981, using advanced methods; a simple adaptation of the original Erdős-Renyi proof was found by Ruciński & Vince (1986), and is presented in Karoński's Handbook chapter. There is another way of defining a random graph \( G \), which is just as natural and common as the model we considered. Rather than choosing the edges of \( G \) independently, we choose the entire graph \( G \) uniformly at random from among all the graphs on \( \{ 0,\ldots, n - 1\} \) that have exactly \( M = M\left( n\right) \) edges: then each of these graphs occurs with the same probability of \( \left( \begin{matrix} N \\ M \end{matrix}\right) \) , where \( N \mathrel{\text{:=}} \left( \begin{array}{l} n \\ 2 \end{array}\right) \) . Just as we studied the likely properties of the graphs in \( \mathcal{G}\left( {n, p}\right) \) for different functions \( p = p\left( n\right) \), we may investigate how the properties of \( G \) in the other model depend on the function \( M\left( n\right) \) . If \( M \) is close to \( {pN} \), the expected number of edges of a graph in \( \mathcal{G}\left( {n, p}\right) \), then the two models behave very similarly. It is then largely a matter of convenience which of them to consider; see Bollobás for details. In order to study threshold phenomena in more detail, one often considers the following random graph process: starting with a \( \overline{{K}^{n}} \) as stage zero, one chooses additional edges one by one (uniformly at random) until the graph is complete. This is a simple example of a Markov chain, whose \( M \) th stage corresponds to the 'uniform' random graph model described above. A survey about threshold phenomena in this setting is given by T. Luczak, The phase transition in a random graph, in (D. Miklós, V.T. Sós & T. Szőnyi, eds.) Paul Erdős is 80, Vol. 2, Proc. Colloq. Math. Soc. János Bolyai (1996). 12 ## Minors Trees and WQO Our goal in this last chapter is a single theorem, one which dwarfs any other result in graph theory and may doubtless be counted among the deepest theorems that mathematics has to offer: in every infinite set of graphs there are two such that one is a minor of the other. This graph minor theorem (or minor theorem for short), inconspicuous though it may look at first glance, has made a fundamental impact both outside graph theory and within. Its proof, due to Neil Robertson and Paul Seymour, takes well over 500 pages. So we have to be modest: of the actual proof of the minor theorem, this chapter will convey only a very rough impression. However, as with most truly fundamental results, the proof has sparked off the development of methods of quite independent interest and potential. This is true particularly for the use of tree-decompositions, a technique we shall meet in Section 12.3. Section 12.1 gives an introduction to well-quasi-ordering, a concept central to the minor theorem. In Section 12.2 we apply this concept to prove the minor theorem for trees. In Section 12.4 we look at forbidden-minor theorems: results in the spirit of Kuratowski's theorem (4.4.6) or Wagner's theorem (7.3.4), which describe the structure of the graphs not containing some specified graph or graphs as a minor. We prove one such theorem in full (excluding a given planar graph) and state another (excluding \( {K}^{n} \) ); both are central results and tools in the theory of graph minors. In Section 12.5 we give a direct proof of the 'generalized Kuratowski' theorem that embeddability in any fixed surface can be characterized by forbidding finitely many minors. We conclude with an overview of the proof and implications of the graph minor theorem itself. ## 12.1 Well-quasi-ordering --- well-quasi-ordering --- A reflexive and transitive relation is called a quasi-ordering. A quasi-ordering \( \leq \) on \( X \) is a well-quasi-ordering, and the elements of \( X \) are well-quasi-ordered by \( \leq \), if for every infinite sequence \( {x}_{0},{x}_{1},\ldots \) in \( X \) good pair there are indices \( i < j \) such that \( {x}_{i} \leq {x}_{j} \) . Then \( \left( {{x}_{i},{x}_{j}}\right) \) is a good pair --- good/bad sequence --- of this sequence. A sequence containing a good pair is a good sequence; thus, a quasi-ordering on \( X \) is a well-quasi-ordering if and only if every infinite sequence in \( X \) is good. An infinite sequence is bad if it is not good. Proposition 12.1.1. A quasi-ordering \( \leq \) on \( X \) is a well-quasi-ordering if and only if \( X \) contains neither an infinite antichain nor an infinite strictly decreasing sequence \( {x}_{0} > {x}_{1} > \ldots \) \( \left( {9.1.2}\right) \) Proof. The forward implication is trivial. Conversely, let \( {x}_{0},{x}_{1},\ldots \) be any infinite sequence in \( X \) . Let \( K \) be the complete graph on \( \mathbb{N} = \) \( \{ 0,1,\ldots \} \) . Colour the edges \( {ij}\left( {i < j}\right) \) of \( K \) with three colours: green if \( {x}_{i} \leq {x}_{j} \), red if \( {x}_{i} > {x}_{j} \), and amber if \( {x}_{i},{x}_{j} \) are incomparable. By Ramsey’s theorem (9.1.2), \( K \) has an infinite induced subgraph whose edges all have the same colour. If there is neither an infinite antichain nor an infinite strictly decreasing sequence in \( X \), then this colour must be green, i.e. \( {x}_{0},{x}_{1},\ldots \) has an infinite subs

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nfinite sequence is bad if it is not good. Proposition 12.1.1. A quasi-ordering \( \leq \) on \( X \) is a well-quasi-ordering if and only if \( X \) contains neither an infinite antichain nor an infinite strictly decreasing sequence \( {x}_{0} > {x}_{1} > \ldots \) \( \left( {9.1.2}\right) \) Proof. The forward implication is trivial. Conversely, let \( {x}_{0},{x}_{1},\ldots \) be any infinite sequence in \( X \) . Let \( K \) be the complete graph on \( \mathbb{N} = \) \( \{ 0,1,\ldots \} \) . Colour the edges \( {ij}\left( {i < j}\right) \) of \( K \) with three colours: green if \( {x}_{i} \leq {x}_{j} \), red if \( {x}_{i} > {x}_{j} \), and amber if \( {x}_{i},{x}_{j} \) are incomparable. By Ramsey’s theorem (9.1.2), \( K \) has an infinite induced subgraph whose edges all have the same colour. If there is neither an infinite antichain nor an infinite strictly decreasing sequence in \( X \), then this colour must be green, i.e. \( {x}_{0},{x}_{1},\ldots \) has an infinite subsequence in which every pair is good. In particular, the sequence \( {x}_{0},{x}_{1},\ldots \) is good. In the proof of Proposition 12.1.1, we showed more than was needed: rather than finding a single good pair in \( {x}_{0},{x}_{1},\ldots \), we found an infinite increasing subsequence. We have thus shown the following: Corollary 12.1.2. If \( X \) is well-quasi-ordered, then every infinite sequence in \( X \) has an infinite increasing subsequence. The following lemma, and the idea of its proof, are fundamental to the theory of well-quasi-ordering. Let \( \leq \) be a quasi-ordering on a set \( X \) . \( \leq \) For finite subsets \( A, B \subseteq X \), write \( A \leq B \) if there is an injective mapping \( f : A \rightarrow B \) such that \( a \leq f\left( a\right) \) for all \( a \in A \) . This naturally extends \( \leq \) to \( \left\lbrack X\right\rbrack < \omega \) a quasi-ordering on \( {\left\lbrack X\right\rbrack }^{ < \omega } \), the set of all finite subsets of \( X \) . \( \left\lbrack \begin{array}{l} {12.2.1} \\ {12.5.1} \end{array}\right\rbrack \) Lemma 12.1.3. If \( X \) is well-quasi-ordered by \( \leq \), then so is \( {\left\lbrack X\right\rbrack }^{ < \omega } \) . Proof. Suppose that \( \leq \) is a well-quasi-ordering on \( X \) but not on \( {\left\lbrack X\right\rbrack }^{ < \omega } \) . We start by constructing a bad sequence \( {\left( {A}_{n}\right) }_{n \in \mathbb{N}} \) in \( {\left\lbrack X\right\rbrack }^{ < \omega } \), as follows. Given \( n \in \mathbb{N} \), assume inductively that \( {A}_{i} \) has been defined for every \( i < n \), and that there exists a bad sequence in \( {\left\lbrack X\right\rbrack }^{ < \omega } \) starting with \( {A}_{0},\ldots ,{A}_{n - 1} \) . (This is clearly true for \( n = 0 \) : by assumption, \( {\left\lbrack X\right\rbrack }^{ < \omega } \) contains a bad sequence, and this has the empty sequence as an initial segment.) Choose \( {A}_{n} \in {\left\lbrack X\right\rbrack }^{ < \omega } \) so that some bad sequence in \( {\left\lbrack X\right\rbrack }^{ < \omega } \) starts with \( {A}_{0},\ldots ,{A}_{n} \) and \( \left| {A}_{n}\right| \) is as small as possible. Clearly, \( {\left( {A}_{n}\right) }_{n \in \mathbb{N}} \) is a bad sequence in \( {\left\lbrack X\right\rbrack }^{ < \omega } \) ; in particular, \( {A}_{n} \neq \varnothing \) for all \( n \) . For each \( n \) pick an element \( {a}_{n} \in {A}_{n} \) and set \( {B}_{n} \mathrel{\text{:=}} {A}_{n} \smallsetminus \left\{ {a}_{n}\right\} \) . By Corollary 12.1.2, the sequence \( {\left( {a}_{n}\right) }_{n \in \mathbb{N}} \) has an infinite increasing subsequence \( {\left( {a}_{{n}_{i}}\right) }_{i \in \mathbb{N}} \) . By the minimal choice of \( {A}_{{n}_{0}} \), the sequence \[ {A}_{0},\ldots ,{A}_{{n}_{0} - 1},{B}_{{n}_{0}},{B}_{{n}_{1}},{B}_{{n}_{2}},\ldots \] is good; consider a good pair. Since \( {\left( {A}_{n}\right) }_{n \in \mathbb{N}} \) is bad, this pair cannot have the form \( \left( {{A}_{i},{A}_{j}}\right) \) or \( \left( {{A}_{i},{B}_{j}}\right) \), as \( {B}_{j} \leq {A}_{j} \) . So it has the form \( \left( {{B}_{i},{B}_{j}}\right) \) . Extending the injection \( {B}_{i} \rightarrow {B}_{j} \) by \( {a}_{i} \mapsto {a}_{j} \), we deduce again that \( \left( {{A}_{i},{A}_{j}}\right) \) is good, a contradiction. ## 12.2 The graph minor theorem for trees The minor theorem can be expressed by saying that the finite graphs are well-quasi-ordered by the minor relation \( \preccurlyeq \) . Indeed, by Proposition 12.1.1 and the obvious fact that no strictly descending sequence of minors can be infinite, being well-quasi-ordered is equivalent to the non-existence of an infinite antichain, the formulation used earlier. In this section, we prove a strong version of the graph minor theorem for trees: ## Theorem 12.2.1. (Kruskal 1960) \( \left\lbrack {12.5.1}\right\rbrack \) The finite trees are well-quasi-ordered by the topological minor relation. We shall base the proof of Theorem 12.2.1 on the following notion of an embedding between rooted trees, which strengthens the usual embedding as a topological minor. Consider two trees \( T \) and \( {T}^{\prime } \), with roots \( r \) and \( {r}^{\prime } \) say. Let us write \( T \leq {T}^{\prime } \) if there exists an isomorphism \( \varphi \), from \( T \leq {T}^{\prime } \) some subdivision of \( T \) to a subtree \( {T}^{\prime \prime } \) of \( {T}^{\prime } \), that preserves the tree-order on \( V\left( T\right) \) associated with \( T \) and \( r \) . (Thus if \( x < y \) in \( T \) then \( \varphi \left( x\right) < \varphi \left( y\right) \) in \( {T}^{\prime } \) ; see Fig. 12.2.1.) As one easily checks, this is a quasi-ordering on the class of all rooted trees. Proof of Theorem 12.2.1. We show that the rooted trees are well- \( \left( {12.1.3}\right) \) quasi-ordered by the relation \( \leq \) defined above; this clearly implies the theorem. Suppose not. To derive a contradiction, we proceed as in the proof of Lemma 12.1.3. Given \( n \in \mathbb{N} \), assume inductively that we have chosen a sequence \( {T}_{0},\ldots ,{T}_{n - 1} \) of rooted trees such that some bad sequence of rooted trees starts with this sequence. Choose as \( {T}_{n} \) a minimum-order  Fig. 12.2.1. An embedding of \( T \) in \( {T}^{\prime } \) showing that \( T \leq {T}^{\prime } \) rooted tree such that some bad sequence starts with \( {T}_{0},\ldots ,{T}_{n} \) . For each \( n \in \mathbb{N} \), denote the root of \( {T}_{n} \) by \( {r}_{n} \) . Clearly, \( {\left( {T}_{n}\right) }_{n \in \mathbb{N}} \) is a bad sequence. For each \( n \), let \( {A}_{n} \) denote the set of components of \( {T}_{n} - {r}_{n} \), made into rooted trees by choosing the neighbours of \( {r}_{n} \) as their roots. Note that the tree-order of these trees is that induced by \( {T}_{n} \) . Let us prove that the set \( A \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{n \in \mathbb{N}}}{A}_{n} \) of all these trees is well-quasi-ordered. Let \( {\left( {T}^{k}\right) }_{k \in \mathbb{N}} \) be any sequence of trees in \( A \) . For every \( k \in \mathbb{N} \) choose an \( n = n\left( k\right) \) such that \( {T}^{k} \in {A}_{n} \) . Pick a \( k \) with smallest \( n\left( k\right) \) . Then \[ {T}_{0},\ldots ,{T}_{n\left( k\right) - 1},{T}^{k},{T}^{k + 1},\ldots \] is a good sequence, by the minimal choice of \( {T}_{n\left( k\right) } \) and \( {T}^{k} \subsetneqq {T}_{n\left( k\right) } \) . Let \( \left( {T,{T}^{\prime }}\right) \) be a good pair of this sequence. Since \( {\left( {T}_{n}\right) }_{n \in \mathbb{N}} \) is bad, \( T \) cannot be among the first \( n\left( k\right) \) members \( {T}_{0},\ldots ,{T}_{n\left( k\right) - 1} \) of our sequence: then \( {T}^{\prime } \) would be some \( {T}^{i} \) with \( i \geq k \), i.e. \[ T \leq {T}^{\prime } = {T}^{i} \leq {T}_{n\left( i\right) } \] since \( n\left( k\right) \leq n\left( i\right) \) by the choice of \( k \), this would make \( \left( {T,{T}_{n\left( i\right) }}\right) \) a good pair in the bad sequence \( {\left( {T}_{n}\right) }_{n \in \mathbb{N}} \) . Hence \( \left( {T,{T}^{\prime }}\right) \) is a good pair also in \( {\left( {T}^{k}\right) }_{k \in \mathbb{N}} \), completing the proof that \( A \) is well-quasi-ordered. By Lemma 12.1.3, \( {}^{1} \) the sequence \( {\left( {A}_{n}\right) }_{n \in \mathbb{N}} \) in \( {\left\lbrack A\right\rbrack }^{ < \omega } \) has a good pair \( \left( {{A}_{i},{A}_{j}}\right) \) ; let \( f : {A}_{i} \rightarrow {A}_{j} \) be injective with \( T \leq f\left( T\right) \) for all \( T \in {A}_{i} \) . Now extend the union of the embeddings \( T \rightarrow f\left( T\right) \) to a map \( \varphi \) from \( V\left( {T}_{i}\right) \) to \( V\left( {T}_{j}\right) \) by letting \( \varphi \left( {r}_{i}\right) \mathrel{\text{:=}} {r}_{j} \) . This map \( \varphi \) preserves the tree-order of \( {T}_{i} \), and it defines an embedding to show that \( {T}_{i} \leq {T}_{j} \), since the edges \( {r}_{i}r \in {T}_{i} \) map naturally to the paths \( {r}_{j}{T}_{j}\varphi \left( r\right) \) . Hence \( \left( {{T}_{i},{T}_{j}}\right) \) is a good pair in our original bad sequence of rooted trees, a contradiction. --- 1 Any readers worried that we might need the lemma for sequences or multisets rather than just sets here, note that isomorphic elements of \( {A}_{n} \) are not identified: we always have \( \left| {A}_{n}\right| = d\left( {r}_{n}\right) \) . --- ## 12.3 Tree-decompositions Trees are graphs with some very distinctive and fundamental properties; consider Theorem 1.5.1 and Corollary 1.5.2, or the more sophisticated example of Kruskal's theorem. It is therefore legitimate to ask to what degree those properties can be transferred to more general graphs, graphs that are not themselves trees but tree-like in some sense. \( {}^{

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varphi \) preserves the tree-order of \( {T}_{i} \), and it defines an embedding to show that \( {T}_{i} \leq {T}_{j} \), since the edges \( {r}_{i}r \in {T}_{i} \) map naturally to the paths \( {r}_{j}{T}_{j}\varphi \left( r\right) \) . Hence \( \left( {{T}_{i},{T}_{j}}\right) \) is a good pair in our original bad sequence of rooted trees, a contradiction. --- 1 Any readers worried that we might need the lemma for sequences or multisets rather than just sets here, note that isomorphic elements of \( {A}_{n} \) are not identified: we always have \( \left| {A}_{n}\right| = d\left( {r}_{n}\right) \) . --- ## 12.3 Tree-decompositions Trees are graphs with some very distinctive and fundamental properties; consider Theorem 1.5.1 and Corollary 1.5.2, or the more sophisticated example of Kruskal's theorem. It is therefore legitimate to ask to what degree those properties can be transferred to more general graphs, graphs that are not themselves trees but tree-like in some sense. \( {}^{2} \) In this section, we study a concept of tree-likeness that permits generalizations of all the tree properties referred to above (including Kruskal's theorem), and which plays a crucial role in the proof of the graph minor theorem. Let \( G \) be a graph, \( T \) a tree, and let \( \mathcal{V} = {\left( {V}_{t}\right) }_{t \in T} \) be a family of vertex sets \( {V}_{t} \subseteq V\left( G\right) \) indexed by the vertices \( t \) of \( T \) . The pair \( \left( {T,\mathcal{V}}\right) \) is called a tree-decomposition of \( G \) if it satisfies the following three conditions: tree- decomposition (T1) \( V\left( G\right) = \mathop{\bigcup }\limits_{{t \in T}}{V}_{t} \) (T2) for every edge \( e \in G \) there exists a \( t \in T \) such that both ends of \( e \) lie in \( {V}_{t} \) ; (T3) \( {V}_{{t}_{1}} \cap {V}_{{t}_{3}} \subseteq {V}_{{t}_{2}} \) whenever \( {t}_{1},{t}_{2},{t}_{3} \in T \) satisfy \( {t}_{2} \in {t}_{1}T{t}_{3} \) . Conditions (T1) and (T2) together say that \( G \) is the union of the subgraphs \( G\left\lbrack {V}_{t}\right\rbrack \) ; we call these subgraphs and the sets \( {V}_{t} \) themselves the parts of \( \left( {T,\mathcal{V}}\right) \) and say that \( \left( {T,\mathcal{V}}\right) \) is a tree-decomposition of \( G \) into these parts parts. Condition (T3) implies that the parts of \( \left( {T,\mathcal{V}}\right) \) are organized into roughly like a tree (Fig. 12.3.1).  Fig. 12.3.1. Edges and parts ruled out by (T2) and (T3) Before we discuss the role that tree-decompositions play in the proof of the minor theorem, let us note some of their basic properties. Consider a fixed tree-decomposition \( \left( {T,\mathcal{V}}\right) \) of \( G \), with \( \mathcal{V} = {\left( {V}_{t}\right) }_{t \in T} \) as above. \( \left( {T,\mathcal{V}}\right) ,{V}_{t} \) Perhaps the most important feature of a tree-decomposition is that it transfers the separation properties of its tree to the graph decomposed: --- 2 What exactly this 'sense' should be will depend both on the property considered and on its intended application. --- Lemma 12.3.1. Let \( {t}_{1}{t}_{2} \) be any edge of \( T \) and let \( {T}_{1},{T}_{2} \) be the components of \( T - {t}_{1}{t}_{2} \), with \( {t}_{1} \in {T}_{1} \) and \( {t}_{2} \in {T}_{2} \) . Then \( {V}_{{t}_{1}} \cap {V}_{{t}_{2}} \) separates \( {U}_{1} \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{t \in {T}_{1}}}{V}_{t} \) from \( {U}_{2} \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{t \in {T}_{2}}}{V}_{t} \) in \( G \) (Fig. 12.3.2).  Fig. 12.3.2. \( {V}_{{t}_{1}} \cap {V}_{{t}_{2}} \) separates \( {U}_{1} \) from \( {U}_{2} \) in \( G \) Proof. Both \( {t}_{1} \) and \( {t}_{2} \) lie on every \( t - {t}^{\prime } \) path in \( T \) with \( t \in {T}_{1} \) and \( {t}^{\prime } \in {T}_{2} \) . Therefore \( {U}_{1} \cap {U}_{2} \subseteq {V}_{{t}_{1}} \cap {V}_{{t}_{2}} \) by (T3), so all we have to show is that \( G \) has no edge \( {u}_{1}{u}_{2} \) with \( {u}_{1} \in {U}_{1} \smallsetminus {U}_{2} \) and \( {u}_{2} \in {U}_{2} \smallsetminus {U}_{1} \) . If \( {u}_{1}{u}_{2} \) is such an edge, then by (T2) there is a \( t \in T \) with \( {u}_{1},{u}_{2} \in {V}_{t} \) . By the choice of \( {u}_{1} \) and \( {u}_{2} \) we have neither \( t \in {T}_{2} \) nor \( t \in {T}_{1} \), a contradiction. Note that tree-decompositions are passed on to subgraphs: \( \left\lbrack {12.4.2}\right\rbrack \) Lemma 12.3.2. For every \( H \subseteq G \), the pair \( \left( {T,{\left( {V}_{t} \cap V\left( H\right) \right) }_{t \in T}}\right) \) is a tree-decomposition of \( H \) . Similarly for contractions: Lemma 12.3.3. Suppose that \( G \) is an \( {MH} \) with branch sets \( {U}_{h} \) , \( h \in V\left( H\right) \) . Let \( f : V\left( G\right) \rightarrow V\left( H\right) \) be the map assigning to each vertex of \( G \) the index of the branch set containing it. For all \( t \in T \) let \( {W}_{t} \mathrel{\text{:=}} \left\{ {f\left( v\right) \mid v \in {V}_{t}}\right\} \), and put \( \mathcal{W} \mathrel{\text{:=}} {\left( {W}_{t}\right) }_{t \in T} \) . Then \( \left( {T,\mathcal{W}}\right) \) is a tree-decomposition of \( H \) . Proof. The assertions (T1) and (T2) for \( \left( {T,\mathcal{W}}\right) \) follow immediately from the corresponding assertions for \( \left( {T,\mathcal{V}}\right) \) . Now let \( {t}_{1},{t}_{2},{t}_{3} \in T \) be as in (T3), and consider a vertex \( h \in {W}_{{t}_{1}} \cap {W}_{{t}_{3}} \) of \( H \) ; we show that \( h \in {W}_{{t}_{2}} \) . By definition of \( {W}_{{t}_{1}} \) and \( {W}_{{t}_{3}} \), there are vertices \( {v}_{1} \in {V}_{{t}_{1}} \cap {U}_{h} \) and \( {v}_{3} \in {V}_{{t}_{3}} \cap {U}_{h} \) . Since \( {U}_{h} \) is connected in \( G \) and \( {V}_{{t}_{2}} \) separates \( {v}_{1} \) from \( {v}_{3} \) in \( G \) by Lemma 12.3.1, \( {V}_{{t}_{2}} \) has a vertex in \( {U}_{h} \) . By definition of \( {W}_{{t}_{2}} \) , this implies \( h \in {W}_{{t}_{2}} \) . Lemma 12.3.4. Given a set \( W \subseteq V\left( G\right) \), there is either a \( t \in T \) such that \( W \subseteq {V}_{t} \), or there are vertices \( {w}_{1},{w}_{2} \in W \) and an edge \( {t}_{1}{t}_{2} \in T \) such that \( {w}_{1},{w}_{2} \) lie outside the set \( {V}_{{t}_{1}} \cap {V}_{{t}_{2}} \) and are separated by it in \( G \) . Proof. Let us orient the edges of \( T \) as follows. For each edge \( {t}_{1}{t}_{2} \in T \) , define \( {U}_{1},{U}_{2} \) as in Lemma 12.3.1; then \( {V}_{{t}_{1}} \cap {V}_{{t}_{2}} \) separates \( {U}_{1} \) from \( {U}_{2} \) . If \( {V}_{{t}_{1}} \cap {V}_{{t}_{2}} \) does not separate any two vertices of \( W \) that lie outside it, we can find an \( i \in \{ 1,2\} \) such that \( W \subseteq {U}_{i} \), and orient \( {t}_{1}{t}_{2} \) towards \( {t}_{i} \) . Let \( t \) be the last vertex of a maximal directed path in \( T \) ; we claim that \( W \subseteq {V}_{t} \) . Given \( w \in W \), let \( {t}^{\prime } \in T \) be such that \( w \in {V}_{{t}^{\prime }} \) . If \( {t}^{\prime } \neq t \) , then the edge \( e \) at \( t \) that separates \( {t}^{\prime } \) from \( t \) in \( T \) is directed towards \( t \) , so \( w \) also lies in \( {V}_{{t}^{\prime \prime }} \) for some \( {t}^{\prime \prime } \) in the component of \( T - e \) containing \( t \) . Therefore \( w \in {V}_{t} \) by (T3). The following special case of Lemma 12.3.4 is used particularly often: Lemma 12.3.5. Any complete subgraph of \( G \) is contained in some part \( \left\lbrack {12.4.2}\right\rbrack \) of \( \left( {T,\mathcal{V}}\right) \) . As indicated by Figure 12.3.1, the parts of \( \left( {T,\mathcal{V}}\right) \) reflect the structure of the tree \( T \), so in this sense the graph \( G \) decomposed resembles a tree. However, this is valuable only inasmuch as the structure of \( G \) within each part is negligible: the smaller the parts, the closer the resemblance. This observation motivates the following definition. The width of width \( \left( {T,\mathcal{V}}\right) \) is the number \[ \max \left\{ {\left| {V}_{t}\right| - 1 : t \in T}\right\} \] and the tree-width \( \operatorname{tw}\left( G\right) \) of \( G \) is the least width of any tree-decomposi- --- tree-width \( \operatorname{tw}\left( G\right) \) --- tion of \( G \) . As one easily checks, \( {}^{3} \) trees themselves have tree-width 1 . By Lemmas 12.3.2 and 12.3.3, the tree-width of a graph will never be increased by deletion or contraction: Proposition 12.3.6. If \( H \preccurlyeq G \) then \( \operatorname{tw}\left( H\right) \leq \operatorname{tw}\left( G\right) \) . \( \left\lbrack {12.4.2}\right\rbrack \) Graphs of bounded tree-width are sufficiently similar to trees that it becomes possible to adapt the proof of Kruskal's theorem to the class of these graphs; very roughly, one has to iterate the 'minimal bad sequence' argument from the proof of Lemma 12.1.3 \( \operatorname{tw}\left( G\right) \) times. This takes us a step further towards a proof of the graph minor theorem: Theorem 12.3.7. (Robertson & Seymour 1990) \( \left\lbrack \begin{array}{l} {12.5.1} \\ {12.5.3} \end{array}\right\rbrack \) For every integer \( k > 0 \), the graphs of tree-width \( < k \) are well-quasi-ordered by the minor relation. --- 3 Indeed the '-1' in the definition of width serves no other purpose than to make this statement true. --- In order to make use of Theorem 12.3.7 for a proof of the general minor theorem, we should be able to say something about the graphs it does not cover, i.e. to deduce some information about a graph from the assumption that its tree-width is large. Our next theorem achieves just that: it identifies a canonical obstruction to small tree-width, a structural phenomenon that occurs in a

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s theorem to the class of these graphs; very roughly, one has to iterate the 'minimal bad sequence' argument from the proof of Lemma 12.1.3 \( \operatorname{tw}\left( G\right) \) times. This takes us a step further towards a proof of the graph minor theorem: Theorem 12.3.7. (Robertson & Seymour 1990) \( \left\lbrack \begin{array}{l} {12.5.1} \\ {12.5.3} \end{array}\right\rbrack \) For every integer \( k > 0 \), the graphs of tree-width \( < k \) are well-quasi-ordered by the minor relation. --- 3 Indeed the '-1' in the definition of width serves no other purpose than to make this statement true. --- In order to make use of Theorem 12.3.7 for a proof of the general minor theorem, we should be able to say something about the graphs it does not cover, i.e. to deduce some information about a graph from the assumption that its tree-width is large. Our next theorem achieves just that: it identifies a canonical obstruction to small tree-width, a structural phenomenon that occurs in a graph if and only if its tree-width is large. touch Let us say that two subsets of \( V\left( G\right) \) touch if they have a vertex in common or \( G \) contains an edge between them. A set of mutually touching bramble connected vertex sets in \( G \) is a bramble. Extending our terminology of cover Chapter 2, we say that a subset of \( V\left( G\right) \) covers (or is a cover of) a bramble \( \mathcal{B} \) if it meets every element of \( \mathcal{B} \) . The least number of vertices order covering a bramble is the order of that bramble. The following simple observation will be useful: Lemma 12.3.8. Any set of vertices separating two covers of a bramble also covers that bramble. Proof. Since each set in the bramble is connected and meets both of the covers, it also meets any set separating these covers. A typical example of a bramble is the set of crosses in a grid. The grid \( k \times k \) grid is the graph on \( \{ 1,\ldots, k{\} }^{2} \) with the edge set \[ \left\{ {\left( {i, j}\right) \left( {{i}^{\prime },{j}^{\prime }}\right) : \left| {i - {i}^{\prime }}\right| + \left| {j - {j}^{\prime }}\right| = 1}\right\} . \] The crosses of this grid are the \( {k}^{2} \) sets \[ {C}_{ij} \mathrel{\text{:=}} \{ \left( {i,\ell }\right) \mid \ell = 1,\ldots, k\} \cup \{ \left( {\ell, j}\right) \mid \ell = 1,\ldots, k\} . \] Thus, the cross \( {C}_{ij} \) is the union of the grid’s \( i \) th column and its \( j \) th row. Clearly, the crosses of the \( k \times k \) grid form a bramble of order \( k \) : they are covered by any row or column, while any set of fewer than \( k \) vertices misses both a row and a column, and hence a cross. The following result is sometimes called the tree-width duality theorem: Theorem 12.3.9. (Seymour & Thomas 1993) Let \( k \geq 0 \) be an integer. A graph has tree-width \( \geq k \) if and only if it contains a bramble of order \( > k \) . (3.3.1) Proof. For the backward implication, let \( \mathcal{B} \) be any bramble in a graph \( G \) . We show that every tree-decomposition \( \left( {T,{\left( {V}_{t}\right) }_{t \in T}}\right) \) of \( G \) has a part that covers \( \mathcal{B} \) . As in the proof of Lemma 12.3.4 we start by orienting the edges \( {t}_{1}{t}_{2} \) of \( T \) . If \( X \mathrel{\text{:=}} {V}_{{t}_{1}} \cap {V}_{{t}_{2}} \) covers \( \mathcal{B} \), we are done. If not, then for each \( B \in \mathcal{B} \) disjoint from \( X \) there is an \( i \in \{ 1,2\} \) such that \( B \subseteq {U}_{i} \smallsetminus X \) (defined as in Lemma 12.3.1); recall that \( B \) is connected. This \( i \) is the same for all such \( B \), because they touch. We now orient the edge \( {t}_{1}{t}_{2} \) towards \( {t}_{i} \) . If every edge of \( T \) is oriented in this way and \( t \) is the last vertex of a maximal directed path in \( T \), then \( {V}_{t} \) covers \( \mathcal{B} \) -just as in the proof of Lemma 12.3.4. To prove the forward direction, we now assume that \( G \) contains no bramble of order \( > k \) . We show that for every bramble \( \mathcal{B} \) in \( G \) there is a \( \mathcal{B} \) -admissible tree-decomposition of \( G \), one in which any part of order B- admissible \( > k \) fails to cover \( \mathcal{B} \) . For \( \mathcal{B} = \varnothing \) this implies that \( \operatorname{tw}\left( G\right) < k \), because every set covers the empty bramble. Let \( \mathcal{B} \) be given, and assume inductively that for every bramble \( {\mathcal{B}}^{\prime } \) containing more sets than \( \mathcal{B} \) there is a \( {\mathcal{B}}^{\prime } \) -admissible tree-decomposition of \( G \) . (The induction starts, since no bramble in \( G \) has more than \( {2}^{\left| G\right| } \) sets.) Let \( X \subseteq V\left( G\right) \) be a cover of \( \mathcal{B} \) with as few vertices as possible; \( X \) then \( \ell \mathrel{\text{:=}} \left| X\right| \leq k \) is the order of \( \mathcal{B} \) . Our aim is to show the following: For every component \( C \) of \( G - X \) there exists a \( \mathcal{B} \) -admissible (*) tree-decomposition of \( G\left\lbrack {X \cup V\left( C\right) }\right\rbrack \) with \( X \) as a part. Then these tree-decompositions can be combined to a \( \mathcal{B} \) -admissible tree-decomposition of \( G \) by identifying their nodes corresponding to \( X \) . (If \( X = V\left( G\right) \), then the tree-decomposition with \( X \) as its only part is \( \mathcal{B} \) - admissible.) So let \( C \) be a fixed component of \( G - X \), write \( H \mathrel{\text{:=}} G\left\lbrack {X \cup V\left( C\right) }\right\rbrack \) , \( C, H \) and put \( {\mathcal{B}}^{\prime } \mathrel{\text{:=}} \mathcal{B} \cup \{ C\} \) . If \( {\mathcal{B}}^{\prime } \) is not a bramble then \( C \) fails to touch \( {\mathcal{B}}^{\prime } \) some element of \( \mathcal{B} \), and hence \( Y \mathrel{\text{:=}} V\left( C\right) \cup N\left( C\right) \) does not cover \( \mathcal{B} \) . Then the tree-decomposition of \( H \) consisting of the two parts \( X \) and \( Y \) satisfies \( \left( *\right) \) . So we may assume that \( {\mathcal{B}}^{\prime } \) is a bramble. Since \( X \) covers \( \mathcal{B} \), we have \( C \notin \mathcal{B} \) and hence \( \left| {\mathcal{B}}^{\prime }\right| > \left| \mathcal{B}\right| \) . Our induction hypothesis therefore ensures that \( G \) has a \( {\mathcal{B}}^{\prime } \) -admissible tree-decomposition \( \left( {T,{\left( {V}_{t}\right) }_{t \in T}}\right) \) . If \( T,{\left( {V}_{t}\right) }_{t \in T} \) this decomposition is also \( \mathcal{B} \) -admissible, there is nothing more to show. If not, then one of its parts of order \( > k,{V}_{s} \) say, covers \( \mathcal{B} \) . Since no set of fewer than \( \ell \) vertices covers \( \mathcal{B} \), Lemma 12.3.8 implies with Menger’s theorem (3.3.1) that \( {V}_{s} \) and \( X \) are linked by \( \ell \) disjoint paths \( {P}_{1},\ldots ,{P}_{\ell } \) . As \( {V}_{s} \) fails to cover \( {\mathcal{B}}^{\prime } \) and hence lies in \( G - C \), the paths \( {P}_{i} \) meet \( H \) only in their ends \( {x}_{i} \in X \) . For each \( i = 1,\ldots ,\ell \) pick a \( {t}_{i} \in T \) with \( {x}_{i} \in {V}_{{t}_{i}} \), and let \[ {W}_{t} \mathrel{\text{:=}} \left( {{V}_{t} \cap V\left( H\right) }\right) \cup \left\{ {{x}_{i} \mid t \in {sT}{t}_{i}}\right\} \] for all \( t \in T \) (Fig. 12.3.3). Then \( \left( {T,{\left( {W}_{t}\right) }_{t \in T}}\right) \) is the tree-decomposition which \( \left( {T,{\left( {V}_{t}\right) }_{t \in T}}\right) \) induces on \( H \) (cf. Lemma 12.3.2), except that a few \( {x}_{i} \) have been added to some of the parts. Despite these additions, we  Fig. 12.3.3. \( {W}_{t} \) contains \( {x}_{2} \) and \( {x}_{3} \) but not \( {x}_{1};{W}_{{t}^{\prime }} \) contains no \( {x}_{i} \) still have \( \left| {W}_{t}\right| \leq \left| {V}_{t}\right| \) for all \( t \) : for each \( {x}_{i} \in {W}_{t} \smallsetminus {V}_{t} \) we have \( t \in {sT}{t}_{i} \) , so \( {V}_{t} \) contains some other vertex of \( {P}_{i} \) (Lemma 12.3.1); that vertex does not lie in \( {W}_{t} \), because \( {P}_{i} \) meets \( H \) only in \( {x}_{i} \) . Moreover, \( \left( {T,{\left( {W}_{t}\right) }_{t \in T}}\right) \) clearly satisfies (T3), because each \( {x}_{i} \) is added to every part along some path in \( T \) containing \( {t}_{i} \), so it is again a tree-decomposition. As \( {W}_{s} = X \), all that is left to show for \( \left( *\right) \) is that this decomposition is \( \mathcal{B} \) -admissible. Consider any \( {W}_{t} \) of order \( > k \) . Then \( {W}_{t} \) meets \( C \) , because \( \left| X\right| = \ell \leq k \) . Since \( \left( {T,{\left( {V}_{t}\right) }_{t \in T}}\right) \) is \( {\mathcal{B}}^{\prime } \) -admissible and \( \left| {V}_{t}\right| \geq \) \( \left| {W}_{t}\right| > k \), we know that \( {V}_{t} \) fails to meet some \( B \in \mathcal{B} \) ; let us show that \( {W}_{t} \) does not meet this \( B \) either. If it does, it must do so in some \( {x}_{i} \in {W}_{t} \smallsetminus {V}_{t} \) . Then \( B \) is a connected set meeting both \( {V}_{s} \) and \( {V}_{{t}_{i}} \) but not \( {V}_{t} \) . As \( t \in {sT}{t}_{i} \) by definition of \( {W}_{t} \), this contradicts Lemma 12.3.1. Often, Theorem 12.3.9 is stated in terms of the bramble number of a graph, the largest order of any bramble in it. The theorem then says that the tree-width of a graph is exactly one less than its bramble number. How useful even the easy backward direction of Theorem 12.3.9 can be is exemplified once more by our example of the crosses bramble in the \( k \times k \) grid: this bramble has order \( k \), so by the theorem the \( k \times k \) grid has tree-width at least \( k - 1 \) . (Try to show this without the theorem!) In fact, the \( k \times k \) grid has tree-width \( k \) (Exercise 2

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

t} \) fails to meet some \( B \in \mathcal{B} \) ; let us show that \( {W}_{t} \) does not meet this \( B \) either. If it does, it must do so in some \( {x}_{i} \in {W}_{t} \smallsetminus {V}_{t} \) . Then \( B \) is a connected set meeting both \( {V}_{s} \) and \( {V}_{{t}_{i}} \) but not \( {V}_{t} \) . As \( t \in {sT}{t}_{i} \) by definition of \( {W}_{t} \), this contradicts Lemma 12.3.1. Often, Theorem 12.3.9 is stated in terms of the bramble number of a graph, the largest order of any bramble in it. The theorem then says that the tree-width of a graph is exactly one less than its bramble number. How useful even the easy backward direction of Theorem 12.3.9 can be is exemplified once more by our example of the crosses bramble in the \( k \times k \) grid: this bramble has order \( k \), so by the theorem the \( k \times k \) grid has tree-width at least \( k - 1 \) . (Try to show this without the theorem!) In fact, the \( k \times k \) grid has tree-width \( k \) (Exercise 21). But more important than its precise value is the fact that the tree-width of grids tends to infinity with their size. For as we shall see, large grid minors pose another canonical obstruction to small tree-width: not only do large grids (and hence all graphs containing large grids as minors; cf. Proposition 12.3.6) have large tree-width, but conversely every graph of large tree-width has a large grid minor (Theorem 12.4.4). Yet another canonical obstruction to small tree-width is described in Exercise 35. Let us call our tree-decomposition \( \left( {T,\mathcal{V}}\right) \) of \( G \) linked, or lean, \( {}^{4} \) if it linked/lean satisfies the following condition: (T4) Given \( {t}_{1},{t}_{2} \in T \) and vertex sets \( {Z}_{1} \subseteq {V}_{{t}_{1}} \) and \( {Z}_{2} \subseteq {V}_{{t}_{2}} \) such that \( \left| {Z}_{1}\right| = \left| {Z}_{2}\right| = : k \), either \( G \) contains \( k \) disjoint \( {Z}_{1} - {Z}_{2} \) paths or there exists an edge \( t{t}^{\prime } \in {t}_{1}T{t}_{2} \) with \( {V}_{t} \cap {V}_{{t}^{\prime }} < k \) . The 'branches' in a lean tree-decomposition are thus stripped of any bulk not necessary to maintain their connecting qualities: if a branch is thick (i.e. the separators \( {V}_{t} \cap {V}_{{t}^{\prime }} \) along a path in \( T \) are large), then \( G \) is highly connected along this branch. For \( {t}_{1} = {t}_{2} \) ,(T4) says that the parts themselves are no larger than their ’external connectivity’ in \( G \) requires; cf. Lemma 12.4.5 and Exercise 35. In our quest for tree-decompositions into 'small' parts, we now have two criteria to choose between: the global 'worst case' criterion of width, which ensures that \( T \) is nontrivial (unless \( G \) is complete) but says nothing about the tree-likeness of \( G \) among parts other than the largest, and the more subtle local criterion of leanness, which ensures tree-likeness everywhere along \( T \) but might be difficult to achieve except with trivial or near-trivial \( T \) . Surprisingly, though, it is always possible to find a tree-decomposition that is optimal with respect to both criteria at once: Theorem 12.3.10. (Thomas 1990) Every graph \( G \) has a lean tree-decomposition of width \( \operatorname{tw}\left( G\right) \) . There is now a short proof of Theorem 12.3.10; see the notes. The fact that this theorem gives us a useful property of minimum-width tree-decompositions 'for free' has made it a valuable tool wherever tree-decompositions are applied. The tree-decomposition \( \left( {T,\mathcal{V}}\right) \) of \( G \) is called simplicial if all the simplicial separators \( {V}_{{t}_{1}} \cap {V}_{{t}_{2}} \) induce complete subgraphs in \( G \) . This assumption can enable us to lift assertions about the parts of the decomposition to \( G \) itself. For example, if all the parts in a simplicial tree-decomposition of \( G \) are \( k \) -colourable, then so is \( G \) (proof?). The same applies to the property of not containing a \( {K}^{r} \) minor for some fixed \( r \) . Algorithmically, it is easy to obtain a simplicial tree-decomposition of a given graph into irreducible parts. Indeed, all we have to do is split the graph recursively along complete separators; if these are always chosen minimal, then the set of parts obtained will even be unique (Exercise 27). Conversely, if \( G \) can be constructed recursively from a set \( \mathcal{H} \) of graphs by pasting along complete subgraphs, then \( G \) has a simplicial tree-decomposition into elements of \( \mathcal{H} \) . For example, by Wagner’s Theorem 7.3.4, any graph without a \( {K}^{5} \) minor has a supergraph with a simplicial tree-decomposition into plane triangulations and copies of the 4 depending on which of the two dual aspects of (T4) we wish to emphasize Wagner graph \( W \), and similarly for graphs without \( {K}^{4} \) minors (see Proposition 12.4.2). Tree-decompositions may thus lead to intuitive structural characterizations of graph properties. A particularly simple example is the following characterization of chordal graphs: \( \left\lbrack {12.4.2}\right\rbrack \) Proposition 12.3.11. \( G \) is chordal if and only if \( G \) has a tree-decomposition into complete parts. \( \left( {5.5.1}\right) \) Proof. We apply induction on \( \left| G\right| \) . We first assume that \( G \) has a tree-decomposition \( \left( {T,\mathcal{V}}\right) \) such that \( G\left\lbrack {V}_{t}\right\rbrack \) is complete for every \( t \in T \) ; let us choose \( \left( {T,\mathcal{V}}\right) \) with \( \left| T\right| \) minimal. If \( \left| T\right| \leq 1 \), then \( G \) is complete and hence chordal. So let \( {t}_{1}{t}_{2} \in T \) be an edge, and for \( i = 1,2 \) define \( {T}_{i} \) and \( {G}_{i} \mathrel{\text{:=}} G\left\lbrack {U}_{i}\right\rbrack \) as in Lemma 12.3.1. Then \( G = {G}_{1} \cup {G}_{2} \) by (T1) and (T2), and \( V\left( {{G}_{1} \cap {G}_{2}}\right) = {V}_{{t}_{1}} \cap {V}_{{t}_{2}} \) by the lemma; thus, \( {G}_{1} \cap {G}_{2} \) is complete. Since \( \left( {{T}_{i},{\left( {V}_{t}\right) }_{t \in {T}_{i}}}\right) \) is a tree-decomposition of \( {G}_{i} \) into complete parts, both \( {G}_{i} \) are chordal by the induction hypothesis. (By the choice of \( \left( {T,\mathcal{V}}\right) \), neither \( {G}_{i} \) is a subgraph of \( G\left\lbrack {{V}_{{t}_{1}} \cap {V}_{{t}_{2}}}\right\rbrack = {G}_{1} \cap {G}_{2} \), so both \( {G}_{i} \) are indeed smaller than \( G \) .) Since \( {G}_{1} \cap {G}_{2} \) is complete, any induced cycle in \( G \) lies in \( {G}_{1} \) or in \( {G}_{2} \) and hence has a chord, so \( G \) too is chordal. Conversely, assume that \( G \) is chordal. If \( G \) is complete, there is nothing to show. If not then, by Proposition 5.5.1, \( G \) is the union of smaller chordal graphs \( {G}_{1},{G}_{2} \) with \( {G}_{1} \cap {G}_{2} \) complete. By the induction hypothesis, \( {G}_{1} \) and \( {G}_{2} \) have tree-decompositions \( \left( {{T}_{1},{\mathcal{V}}_{1}}\right) \) and \( \left( {{T}_{2},{\mathcal{V}}_{2}}\right) \) into complete parts. By Lemma 12.3.5, \( {G}_{1} \cap {G}_{2} \) lies inside one of those parts in each case, say with indices \( {t}_{1} \in {T}_{1} \) and \( {t}_{2} \in {T}_{2} \) . As one easily checks, \( \left( {\left( {{T}_{1} \cup {T}_{2}}\right) + {t}_{1}{t}_{2},{\mathcal{V}}_{1} \cup {\mathcal{V}}_{2}}\right) \) is a tree-decomposition of \( G \) into complete parts. Corollary 12.3.12. \( \operatorname{tw}\left( G\right) = \min \{ \omega \left( H\right) - 1 \mid G \subseteq H;H \) chordal \( \} \) . Proof. By Lemma 12.3.5 and Proposition 12.3.11, each of the graphs \( H \) considered for the minimum has a tree-decomposition of width \( \omega \left( H\right) - 1 \) . Every such tree-decomposition induced one of \( G \) by Lemma 12.3.2, so \( \operatorname{tw}\left( G\right) \leq \omega \left( H\right) - 1 \) for every \( H \) . Conversely, let us construct an \( H \) as above with \( \omega \left( H\right) - 1 \leq \operatorname{tw}\left( G\right) \) . Let \( \left( {T,\mathcal{V}}\right) \) be a tree-decomposition of \( G \) of width \( \operatorname{tw}\left( G\right) \) . For every \( t \in T \) let \( {K}_{t} \) denote the complete graph on \( {V}_{t} \), and put \( H \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{t \in T}}{K}_{t} \) . Clearly, \( \left( {T,\mathcal{V}}\right) \) is also a tree-decomposition of \( H \) . By Proposition 12.3.11, \( H \) is chordal, and by Lemma 12.3.5, \( \omega \left( H\right) - 1 \) is at most the width of \( \left( {T,\mathcal{V}}\right) \) , i.e. at most \( \operatorname{tw}\left( G\right) \) . ## 12.4 Tree-width and forbidden minors If \( \mathcal{H} \) is any set or class of graphs, then the class \[ {\text{Forb}}_{ \preccurlyeq }\left( \mathcal{H}\right) \mathrel{\text{:=}} \{ \;G \mid G \nsucceq H\text{ for all }H \in \mathcal{H}\;\} \] \( {\operatorname{Forb}}_{ \preccurlyeq }\left( \mathcal{H}\right) \) of all graphs without a minor in \( \mathcal{H} \) is a graph property, i.e. is closed under isomorphism. \( {}^{5} \) When it is written as above, we say that this property is expressed by specifying the graphs \( H \in \mathcal{H} \) as forbidden (or excluded) forbidden minors minors. By Proposition 1.7.3, \( {\operatorname{Forb}}_{ \preccurlyeq }\left( \mathcal{H}\right) \) is closed under taking minors, or \( \left( {1.7.3}\right) \) minor-closed: if \( {G}^{\prime } \preccurlyeq G \in {\operatorname{Forb}}_{ \preccurlyeq }\left( \mathcal{H}\right) \) then \( {G}^{\prime } \in {\operatorname{Forb}}_{ \preccurlyeq }\left( \mathcal{H}\right) \) . Every minor-closed property can in turn be expressed by forbidden minors: Proposition 12.4.1. A graph property \( \mathcal{P} \) can be expressed by forbidden minors if and only if it is closed under taking minors. Proof. For the ’if’ part, note that \( \mathcal{P} = {\operatorname{Forb}}_{ \preccurlyeq }\left( \overline{\mathcal{P}}\r

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

of all graphs without a minor in \( \mathcal{H} \) is a graph property, i.e. is closed under isomorphism. \( {}^{5} \) When it is written as above, we say that this property is expressed by specifying the graphs \( H \in \mathcal{H} \) as forbidden (or excluded) forbidden minors minors. By Proposition 1.7.3, \( {\operatorname{Forb}}_{ \preccurlyeq }\left( \mathcal{H}\right) \) is closed under taking minors, or \( \left( {1.7.3}\right) \) minor-closed: if \( {G}^{\prime } \preccurlyeq G \in {\operatorname{Forb}}_{ \preccurlyeq }\left( \mathcal{H}\right) \) then \( {G}^{\prime } \in {\operatorname{Forb}}_{ \preccurlyeq }\left( \mathcal{H}\right) \) . Every minor-closed property can in turn be expressed by forbidden minors: Proposition 12.4.1. A graph property \( \mathcal{P} \) can be expressed by forbidden minors if and only if it is closed under taking minors. Proof. For the ’if’ part, note that \( \mathcal{P} = {\operatorname{Forb}}_{ \preccurlyeq }\left( \overline{\mathcal{P}}\right) \), where \( \overline{\mathcal{P}} \) is the \( \overline{\mathcal{P}} \) complement of \( \mathcal{P} \) . In Section 12.5, we shall return to the general question of how a given minor-closed property is best represented by forbidden minors. In this section, we are interested in one particular example of such a property: bounded tree-width. Consider the property of having tree-width less than some given integer \( k \) . By Propositions 12.3.6 and 12.4.1, this property can be expressed by forbidden minors. Choosing their set \( \mathcal{H} \) as small as possible, we find that \( \mathcal{H} = \left\{ {K}^{3}\right\} \) for \( k = 2 \) : the graphs of tree-width \( < 2 \) are precisely the forests. For \( k = 3 \), we have \( \mathcal{H} = \left\{ {K}^{4}\right\} \) : Proposition 12.4.2. A graph has tree-width \( < 3 \) if and only if it has no \( {K}^{4} \) minor. (7.3.1) \( \left( {12.3.2}\right) \) Proof. By Lemma 12.3.5, we have \( \operatorname{tw}\left( {K}^{4}\right) \geq 3 \) . By Proposition 12.3.6, (12.3.5) therefore, a graph of tree-width \( < 3 \) cannot contain \( {K}^{4} \) as a minor. (12.3.6) Conversely, let \( G \) be a graph without a \( {K}^{4} \) minor; we assume that (12.3.11) \( \left| G\right| \geq 3 \) . Add edges to \( G \) until the graph \( {G}^{\prime } \) obtained is edge-maximal without a \( {K}^{4} \) minor. By Proposition 7.3.1, \( {G}^{\prime } \) can be constructed recursively from triangles by pasting along \( {K}^{2}\mathrm{\;s} \) . By induction on the number of recursion steps and Lemma 12.3.5, every graph constructible in this way has a tree-decomposition into triangles (as in the proof of Proposition 12.3.11). Such a tree-decomposition of \( {G}^{\prime } \) has width 2, and by Lemma 12.3.2 it is also a tree-decomposition of \( G \) . --- 5 As usual, we abbreviate Forb \( \preccurlyeq \left( {\{ H\} }\right) \) to \( {\operatorname{Forb}}_{ \preccurlyeq }\left( H\right) \) . --- As \( k \) grows, the list of forbidden minors characterizing the graphs of tree-width \( < k \) seems to grow fast. They are known explicitly only up to \( k = 4 \) ; see the notes. A question converse to the above is to ask for which \( H \) (other than \( {K}^{3} \) and \( {K}^{4} \) ) the tree-width of the graphs in \( {\operatorname{Forb}}_{ \preccurlyeq }\left( H\right) \) is bounded. Interestingly, it is not difficult to show that any such \( H \) must be planar. (4.4.6) Indeed, as all grids and their minors are planar (why?), every class \( {\operatorname{Forb}}_{ \preccurlyeq }\left( H\right) \) with non-planar \( H \) contains all grids; yet as we saw after Theorem 12.3.9, the grids have unbounded tree-width. The following deep and surprising theorem says that, conversely, the tree-width of the graphs in \( {\operatorname{Forb}}_{ \preccurlyeq }\left( H\right) \) is bounded for every planar \( H \) : Theorem 12.4.3. (Robertson & Seymour 1986) Given a graph \( H \), the graphs without an \( H \) minor have bounded tree-width if and only if \( H \) is planar. The rest of this section is devoted to the proof of Theorem 12.4.3 and an application. To prove Theorem 12.4.3 we have to show that forbidding any planar graph \( H \) as a minor bounds the tree-width of a graph. In fact, we only have to show this for the special cases when \( H \) is a grid, because every planar graph is a minor of some grid. (To see this, take a drawing of the graph, fatten its vertices and edges, and superimpose a sufficiently fine plane grid.) It thus suffices to show the following: Theorem 12.4.4. (Robertson & Seymour 1986) For every integer \( r \) there is an integer \( k \) such that every graph of tree-width at least \( k \) has an \( r \times r \) grid minor. Our proof of Theorem 12.4.4 proceeds as follows. Let \( r \) be given, and let \( G \) be any graph of large enough tree-width (depending on \( r \) ). We first show that \( G \) contains a large family \( \mathcal{A} = \left\{ {{A}_{1},\ldots ,{A}_{m}}\right\} \) of disjoint connected vertex sets such that each pair \( {A}_{i},{A}_{j} \in \mathcal{A} \) can be linked in \( G \) by a family \( {\mathcal{P}}_{ij} \) of many disjoint \( {A}_{i} - {A}_{j} \) paths avoiding all the other sets in \( \mathcal{A} \) . We then consider all the pairs \( \left( {{\mathcal{P}}_{ij},{\mathcal{P}}_{{i}^{\prime }{j}^{\prime }}}\right) \) of these path families. If we can find a pair among these such that many of the paths in \( {\mathcal{P}}_{ij} \) meet many of the paths in \( {\mathcal{P}}_{{i}^{\prime }{j}^{\prime }} \), we shall think of the paths in \( {\mathcal{P}}_{ij} \) as horizontal and the paths in \( {\mathcal{P}}_{{i}^{\prime }{j}^{\prime }} \) as vertical and extract a subdivision of an \( r \times r \) grid from their union. (This will be the difficult part of the proof, because these paths will in general meet in a less orderly way than they do in a grid.) If not, then for every pair \( \left( {{\mathcal{P}}_{ij},{\mathcal{P}}_{{i}^{\prime }{j}^{\prime }}}\right) \) many of the paths in \( {\mathcal{P}}_{ij} \) avoid many of the paths in \( {\mathcal{P}}_{{i}^{\prime }{j}^{\prime }} \) . We can then select one path \( {P}_{ij} \in {\mathcal{P}}_{ij} \) from each family so that these selected paths are pairwise disjoint. Contracting each of the connected sets \( A \in \mathcal{A} \) will then give us a \( {K}^{m} \) minor in \( G \), which contains the desired \( r \times r \) grid if \( m \geq {r}^{2} \) . To implement these ideas formally, we need a few definitions. Let us call a set \( X \subseteq V\left( G\right) \) externally \( k \) -connected in \( G \) if \( \left| X\right| \geq k \) and for externally \( k \) -connected all disjoint subsets \( Y, Z \subseteq X \) with \( \left| Y\right| = \left| Z\right| \leq k \) there are \( \left| Y\right| \) disjoint \( Y - Z \) paths in \( G \) that have no inner vertex or edge in \( G\left\lbrack X\right\rbrack \) . Note that the vertex set of a \( k \) -connected subgraph of \( G \) need not be externally \( k \) -connected in \( G \) . On the other hand, any horizontal path in the \( r \times r \) grid is externally \( k \) -connected in that grid for every \( k \leq r \) . (How?) One of the first things we shall prove below is that any graph of large enough tree-width - not just grids - contains a large externally \( k \) - connected set of vertices (Lemma 12.4.5). Conversely, it is easy to show that large externally \( k \) -connected sets (with \( k \) large) can exist only in graphs of large tree-width (Exercise 35). So, like large grid minors, these sets form a canonical obstruction to small tree-width: they can be found in a graph if and only if its tree-width is large. An ordered pair \( \left( {A, B}\right) \) of subgraphs of \( G \) will be called a premesh premesh in \( G \) if \( G = A \cup B \) and \( A \) contains a tree \( T \) such that (i) \( T \) has maximum degree \( \leq 3 \) ; (ii) every vertex of \( A \cap B \) lies in \( T \) and has degree \( \leq 2 \) in \( T \) ; (iii) \( T \) has a leaf in \( A \cap B \), or \( \left| T\right| = 1 \) and \( T \subseteq A \cap B \) . The order of such a premesh is the number \( \left| {A \cap B}\right| \), and if \( V\left( {A \cap B}\right) \) is order externally \( k \) -connected in \( B \) then this premesh is a \( k \) -mesh in \( G \) . \( k \) -mesh Lemma 12.4.5. Let \( G \) be a graph and let \( h \geq k \geq 1 \) be integers. If \( G \) contains no \( k \) -mesh of order \( h \) then \( G \) has tree-width \( < h + k - 1 \) . Proof. We may assume that \( G \) is connected. Let \( U \subseteq V\left( G\right) \) be max- (3.3.1) imal such that \( G\left\lbrack U\right\rbrack \) has a tree-decomposition \( \mathcal{D} \) of width \( < h + k - 1 \) \( U \) with the additional property that, for every component \( C \) of \( G - U \), the \( \mathcal{D} \) neighbours of \( C \) in \( U \) lie in one part of \( \mathcal{D} \) and \( \left( {G - C,\widetilde{C}}\right) \) is a premesh of order \( \leq h \), where \( \widetilde{C} \mathrel{\text{:=}} G\left\lbrack {V\left( C\right) \cup N\left( C\right) }\right\rbrack \) . Clearly, \( U \neq \varnothing \) . \( \widetilde{C} \) We claim that \( U = V\left( G\right) \) . Suppose not. Let \( C \) be a component of \( G - U \), put \( X \mathrel{\text{:=}} N\left( C\right) \), and let \( T \) be a tree associated with the premesh \( X \) \( \left( {G - C,\widetilde{C}}\right) \) . \( T \) By assumption, \( \left| X\right| \leq h \) ; let us show that equality holds here. If not, let \( u \in X \) be a leaf of \( T \) (respectively \( \{ u\} \mathrel{\text{:=}} V\left( T\right) \) ) as in (iii), and let \( v \in C \) be a neighbour of \( u \) . Put \( {U}^{\prime } \mathrel{\text{:=}} U \cup \{ v\} \) and \( {X}^{\prime } \mathrel{\text{:=}} X \cup \{ v\} \) , let \( {T}^{\prime } \) be the tree obtained from \( T \) by joining \( v \) to \( u \), and let \( {\mathca

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

\left( {G - C,\widetilde{C}}\right) \) is a premesh of order \( \leq h \), where \( \widetilde{C} \mathrel{\text{:=}} G\left\lbrack {V\left( C\right) \cup N\left( C\right) }\right\rbrack \) . Clearly, \( U \neq \varnothing \) . \( \widetilde{C} \) We claim that \( U = V\left( G\right) \) . Suppose not. Let \( C \) be a component of \( G - U \), put \( X \mathrel{\text{:=}} N\left( C\right) \), and let \( T \) be a tree associated with the premesh \( X \) \( \left( {G - C,\widetilde{C}}\right) \) . \( T \) By assumption, \( \left| X\right| \leq h \) ; let us show that equality holds here. If not, let \( u \in X \) be a leaf of \( T \) (respectively \( \{ u\} \mathrel{\text{:=}} V\left( T\right) \) ) as in (iii), and let \( v \in C \) be a neighbour of \( u \) . Put \( {U}^{\prime } \mathrel{\text{:=}} U \cup \{ v\} \) and \( {X}^{\prime } \mathrel{\text{:=}} X \cup \{ v\} \) , let \( {T}^{\prime } \) be the tree obtained from \( T \) by joining \( v \) to \( u \), and let \( {\mathcal{D}}^{\prime } \) be the tree-decomposition of \( G\left\lbrack {U}^{\prime }\right\rbrack \) obtained from \( \mathcal{D} \) by adding \( {X}^{\prime } \) as a new part (joined to a part of \( \mathcal{D} \) containing \( X \), which exists by our choice of \( U \) ; see Fig. 12.4.1). Clearly \( {\mathcal{D}}^{\prime } \) still has width \( < h + k - 1 \) . Consider a component \( {C}^{\prime } \) of \( G - {U}^{\prime } \) . If \( {C}^{\prime } \cap C = \varnothing \) then \( {C}^{\prime } \) is also a component of \( G - U \), so \( N\left( {C}^{\prime }\right) \) lies inside a part of \( \mathcal{D} \) (and hence of \( {\mathcal{D}}^{\prime } \) ), and \( \left( {G - {C}^{\prime },{\widetilde{C}}^{\prime }}\right) \) is a premesh of order \( \leq h \) by assumption. If \( {C}^{\prime } \cap C \neq \varnothing \), then \( {C}^{\prime } \subseteq C \) and \( N\left( {C}^{\prime }\right) \subseteq {X}^{\prime } \) . Moreover, \( v \in N\left( {C}^{\prime }\right) \) : otherwise \( N\left( {C}^{\prime }\right) \subseteq X \) would separate \( {C}^{\prime } \) from \( v \), contradicting the fact that \( {C}^{\prime } \) and \( v \) lie in the same component \( C \) of \( G - X \) . Since \( v \) is a leaf of \( {T}^{\prime } \), it is straightforward to check that \( \left( {G - {C}^{\prime },{\widetilde{C}}^{\prime }}\right) \) is again a premesh of order \( \leq h \), contrary to the maximality of \( U \) .  Fig. 12.4.1. Extending \( U \) and \( \mathcal{D} \) when \( \left| X\right| < h \) Thus \( \left| X\right| = h \), so by assumption our premesh \( \left( {G - C,\widetilde{C}}\right) \) cannot be a \( k \) -mesh; let \( Y, Z \subseteq X \) be sets to witness this. Let \( \mathcal{P} \) be a set of as many disjoint \( Y - Z \) paths in \( H \mathrel{\text{:=}} G\left\lbrack {V\left( C\right) \cup Y \cup Z}\right\rbrack - E\left( {G\left\lbrack {Y \cup Z}\right\rbrack }\right) \) as possible. Since all these paths are ’external’ to \( X \) in \( \widetilde{C} \), we have \( {k}^{\prime } \) \( {k}^{\prime } \mathrel{\text{:=}} \left| \mathcal{P}\right| < \left| Y\right| = \left| Z\right| \leq k \) by the choice of \( Y \) and \( Z \) . By Menger’s \( S \) theorem (3.3.1), \( Y \) and \( Z \) are separated in \( H \) by a set \( S \) of \( {k}^{\prime } \) vertices. Clearly, \( S \) has exactly one vertex on each path in \( \mathcal{P} \) ; we denote the path \( {P}_{s} \) containing the vertex \( s \in S \) by \( {P}_{s} \) (Fig. 12.4.2).  Fig. 12.4.2. \( S \) separates \( Y \) from \( Z \) in \( H \) Let \( {X}^{\prime } \mathrel{\text{:=}} X \cup S \) and \( {U}^{\prime } \mathrel{\text{:=}} U \cup S \), and let \( {\mathcal{D}}^{\prime } \) be the tree-decomposition of \( G\left\lbrack {U}^{\prime }\right\rbrack \) obtained from \( \mathcal{D} \) by adding \( {X}^{\prime } \) as a new part. Clearly, \( \left| {X}^{\prime }\right| \leq \left| X\right| + \left| S\right| \leq h + k - 1 \) . We show that \( {U}^{\prime } \) contradicts the maximality of \( U \) . Since \( Y \cup Z \subseteq N\left( C\right) \) and \( \left| S\right| < \left| Y\right| = \left| Z\right| \) we have \( S \cap C \neq \varnothing \), so \( {U}^{\prime } \) is larger than \( U \) . Let \( {C}^{\prime } \) be a component of \( G - {U}^{\prime } \) . If \( {C}^{\prime } \cap C = \varnothing \) , we argue as earlier. So \( {C}^{\prime } \subseteq C \) and \( N\left( {C}^{\prime }\right) \subseteq {X}^{\prime } \) . As before, \( {C}^{\prime } \) has at least one neighbour \( v \) in \( S \cap C \), since \( X \) cannot separate \( {C}^{\prime } \subseteq C \) from \( S \cap C \) . By definition of \( S,{C}^{\prime } \) cannot have neighbours in both \( Y \smallsetminus S \) and \( Z \smallsetminus S \) ; we assume it has none in \( Y \smallsetminus S \) . Let \( {T}^{\prime } \) be the union of \( T \) and all the \( Y - S \) subpaths of paths \( {P}_{s} \) with \( s \in N\left( {C}^{\prime }\right) \cap C \) ; since these subpaths start in \( Y \smallsetminus S \) and have no inner vertices in \( {X}^{\prime } \), they cannot meet \( {C}^{\prime } \) . Therefore \( \left( {G - {C}^{\prime },{\widetilde{C}}^{\prime }}\right) \) is a premesh with tree \( {T}^{\prime } \) and leaf \( v \) ; the degree conditions on \( {T}^{\prime } \) are easily checked. Its order is \( \left| {N\left( {C}^{\prime }\right) }\right| \leq \) \( \left| X\right| - \left| Y\right| + \left| S\right| = h - \left| Y\right| + {k}^{\prime } < h \), a contradiction to the maximality of \( U \) . Lemma 12.4.6. Let \( k \geq 2 \) be an integer. Let \( T \) be a tree of maximum degree \( \leq 3 \) and \( X \subseteq V\left( T\right) \) . Then \( T \) has a set \( F \) of edges such that every component of \( T - F \) has between \( k \) and \( {2k} - 1 \) vertices in \( X \), except that one such component may have fewer vertices in \( X \) . Proof. We apply induction on \( \left| X\right| \) . If \( \left| X\right| \leq {2k} - 1 \) we put \( F = \varnothing \) . So assume that \( \left| X\right| \geq {2k} \) . Let \( e \) be an edge of \( T \) such that some component \( {T}^{\prime } \) of \( T - e \) has at least \( k \) vertices in \( X \) and \( \left| {T}^{\prime }\right| \) is as small as possible. As \( \Delta \left( T\right) \leq 3 \), the end of \( e \) in \( {T}^{\prime } \) has degree at most two in \( {T}^{\prime } \), so the minimality of \( {T}^{\prime } \) implies that \( \left| {X \cap V\left( {T}^{\prime }\right) }\right| \leq {2k} - 1 \) . Applying the induction hypothesis to \( T - {T}^{\prime } \) we complete the proof. Lemma 12.4.7. Let \( G \) be a bipartite graph with bipartition \( \{ A, B\} \) , \( \left| A\right| = a,\left| B\right| = b \), and let \( c \leq a \) and \( d \leq b \) be positive integers. Assume that \( G \) has at most \( \left( {a - c}\right) \left( {b - d}\right) /d \) edges. Then there exist \( C \subseteq A \) and \( D \subseteq B \) such that \( \left| C\right| = c \) and \( \left| D\right| = d \) and \( C \cup D \) is independent in \( G \) . Proof. As \( \parallel G\parallel \leq \left( {a - c}\right) \left( {b - d}\right) /d \), fewer than \( b - d \) vertices in \( B \) have more than \( \left( {a - c}\right) /d \) neighbours in \( A \) . Choose \( D \subseteq B \) so that \( \left| D\right| = d \) and each vertex in \( D \) has at most \( \left( {a - c}\right) /d \) neighbours in \( A \) . Then \( D \) sends a total of at most \( a - c \) edges to \( A \), so \( A \) has a subset \( C \) of \( c \) vertices without a neighbour in \( D \) . Given a tree \( T \), call an \( r \) -tuple \( \left( {{x}_{1},\ldots ,{x}_{r}}\right) \) of distinct vertices of \( T \) --- good \( r \) -tuple --- good if, for every \( j = 1,\ldots, r - 1 \), the \( {x}_{j} - {x}_{j + 1} \) path in \( T \) contains none of the other vertices in this \( r \) -tuple. Lemma 12.4.8. Every tree \( T \) of order at least \( r\left( {r - 1}\right) \) contains a good \( r \) -tuple of vertices. Proof. Pick a vertex \( x \in T \) . Then \( T \) is the union of its subpaths \( {xTy} \) , where \( y \) ranges over its leaves. Hence unless one of these paths has at least \( r \) vertices, \( T \) has at least \( \left| T\right| /\left( {r - 1}\right) \geq r \) leaves. Since any path of \( r \) vertices and any set of \( r \) leaves gives rise to a good \( r \) -tuple in \( T \), this proves the assertion. Our next lemma shows how to obtain a grid from two large systems of paths that intersect in a particularly orderly way. Lemma 12.4.9. Let \( d, r \geq 2 \) be integers such that \( d \geq {r}^{{2r} + 2} \) . Let \( G \) be a graph containing a set \( \mathcal{H} \) of \( {r}^{2} - 1 \) disjoint paths and a set \( \mathcal{V} = \left\{ {{V}_{1},\ldots ,{V}_{d}}\right\} \) of \( d \) disjoint paths. Assume that every path in \( \mathcal{V} \) meets every path in \( \mathcal{H} \), and that each path \( H \in \mathcal{H} \) consists of \( d \) consecutive (vertex-disjoint) segments such that \( {V}_{i} \) meets \( H \) only in its \( i \) th segment, for every \( i = 1,\ldots, d \) (Fig. 12.4.3). Then \( G \) has an \( r \times r \) grid minor.  Fig. 12.4.3. Paths intersecting as in Lemma 12.4.9 Proof. For each \( i = 1,\ldots, d \), consider the graph with vertex set \( \mathcal{H} \) in which two paths are adjacent whenever \( {V}_{i} \) contains a subpath between them that meets no other path in \( \mathcal{H} \) . Since \( {V}_{i} \) meets every path in \( \mathcal{H} \), this \( {T}_{i} \) is a connected graph; let \( {T}_{i} \) be a spanning tree in it. Since \( \left| \mathcal{H}\right| \geq r\left( {r - 1}\right) \) , Lemma 12.4.8 imp

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

ht\} \) of \( d \) disjoint paths. Assume that every path in \( \mathcal{V} \) meets every path in \( \mathcal{H} \), and that each path \( H \in \mathcal{H} \) consists of \( d \) consecutive (vertex-disjoint) segments such that \( {V}_{i} \) meets \( H \) only in its \( i \) th segment, for every \( i = 1,\ldots, d \) (Fig. 12.4.3). Then \( G \) has an \( r \times r \) grid minor.  Fig. 12.4.3. Paths intersecting as in Lemma 12.4.9 Proof. For each \( i = 1,\ldots, d \), consider the graph with vertex set \( \mathcal{H} \) in which two paths are adjacent whenever \( {V}_{i} \) contains a subpath between them that meets no other path in \( \mathcal{H} \) . Since \( {V}_{i} \) meets every path in \( \mathcal{H} \), this \( {T}_{i} \) is a connected graph; let \( {T}_{i} \) be a spanning tree in it. Since \( \left| \mathcal{H}\right| \geq r\left( {r - 1}\right) \) , Lemma 12.4.8 implies that each of these \( d \geq {r}^{2}{\left( {r}^{2}\right) }^{r} \) trees \( {T}_{i} \) has a good \( r \) -tuple of vertices. Since there are no more than \( {\left( {r}^{2}\right) }^{r} \) distinct \( r \) -tuples --- \( {H}^{1},\ldots ,{H}^{r} \) \( I,{i}_{k} \) \( {\mathcal{H}}^{\prime } \) --- on \( \mathcal{H} \), some \( {r}^{2} \) of the trees \( {T}_{i} \) have a common good \( r \) -tuple \( \left( {{H}^{1},\ldots ,{H}^{r}}\right) \) . Let \( I = \left\{ {{i}_{1},\ldots ,{i}_{{r}^{2}}}\right\} \) be the index set of these trees (with \( {i}_{j} < {i}_{k} \) for \( j < k) \) and put \( {\mathcal{H}}^{\prime } \mathrel{\text{:=}} \left\{ {{H}^{1},\ldots ,{H}^{r}}\right\} \) . Here is an informal description of how we construct our \( r \times r \) grid. Its ’horizontal’ paths will be the paths \( {H}^{1},\ldots ,{H}^{r} \) . Its ’vertical’ paths will be pieced together edge by edge, as follows. The \( r - 1 \) edges of the first vertical path will come from the first \( r - 1 \) trees \( {T}_{i} \), trees with their index \( i \) among the first \( r \) elements of \( I \) . More precisely, its ’edge’ between \( {H}^{j} \) and \( {H}^{j + 1} \) will be the sequence of subpaths of \( {V}_{{i}_{j}} \) (together with some connecting horizontal bits taken from paths in \( \mathcal{H} \smallsetminus {\mathcal{H}}^{\prime } \) ) induced by the edges of an \( {H}^{j} - {H}^{j + 1} \) path in \( {T}_{{i}_{j}} \) that has no inner vertices in \( {\mathcal{H}}^{\prime } \) ; see Fig. 12.4.4. (This is why we need \( \left( {{H}^{1},\ldots ,{H}^{r}}\right) \) to be a good \( r \) -tuple in every tree \( {T}_{i} \) .) Similarly, the \( j \) th edge of the second vertical path will come from an \( {H}^{j} - {H}^{j + 1} \) path in \( {T}_{{i}_{r + j}} \), and so on. (Although we need only \( r - 1 \) edges for each vertical path, we reserve \( r \) rather than just \( r - 1 \) of the paths \( {V}_{i} \) for each vertical path to make the indexing more lucid.  Fig. 12.4.4. An \( {H}^{j} - {H}^{j + 1} \) path in \( {T}_{{i}_{j}} \) inducing segments of \( {V}_{{i}_{j}} \) for the \( j \) th edge of the grid’s first vertical path The paths \( {V}_{{i}_{r}},{V}_{{i}_{2r}},\ldots \) are left unused.) To merge these individual edges into \( r \) vertical paths, we then contract in each horizontal path the initial segment that meets the first \( r \) paths \( {V}_{i} \) with \( i \in I \), then contract the segment that meets the following \( r \) paths \( {V}_{i} \) with \( i \in I \), and so on. Formally, we proceed as follows. Consider all \( j, k \in \{ 1,\ldots, r\} \) . (We shall think of the index \( j \) as counting the horizontal paths, and of the index \( k \) as counting the vertical paths of the grid to be constructed.) Let \( {H}_{k}^{j} \) be the minimal subpath of \( {H}^{j} \) that contains the \( i \) th segment of \( {H}^{j} \) for all \( i \) with \( {i}_{\left( {k - 1}\right) r} < i \leq {i}_{kr} \) (put \( {i}_{0} \mathrel{\text{:=}} 0 \) ). Let \( {\widehat{H}}^{j} \) be obtained from \( {H}^{j} \) by first deleting any vertices following its \( {i}_{{r}^{2}} \) th segment and then contracting every subpath \( {H}_{k}^{j} \) to one vertex \( {v}_{k}^{j} \) . Thus, \( {\check{H}}^{j} = {v}_{1}^{j}\ldots {v}_{r}^{j} \) . Given \( j \in \{ 1,\ldots, r - 1\} \) and \( k \in \{ 1,\ldots, r\} \), we have to define a path \( {V}_{k}^{j} \) that will form the subdivided ’vertical edge’ \( {v}_{k}^{j}{v}_{k}^{j + 1} \) . This path will consist of segments of the path \( {V}_{i} \) together with some otherwise unused segments of paths from \( \mathcal{H} \smallsetminus {\mathcal{H}}^{\prime } \), for \( i \mathrel{\text{:=}} {i}_{\left( {k - 1}\right) r + j} \) ; recall that, by definition of \( {\widehat{H}}^{j} \) and \( {\widehat{H}}^{j + 1} \), this \( {V}_{i} \) does indeed meet \( {H}^{j} \) and \( {H}^{j + 1} \) precisely in vertices that were contracted into \( {v}_{k}^{j} \) and \( {v}_{k}^{j + 1} \), respectively. To define \( {V}_{k}^{j} \), consider an \( {H}^{j} - {H}^{j + 1} \) path \( P = {H}_{1}\ldots {H}_{t} \) in \( {T}_{i} \) that has no inner vertices in \( {\mathcal{H}}^{\prime } \) . (Thus, \( {H}_{1} = {H}^{j} \) and \( {H}_{t} = {H}^{j + 1} \) .) Every edge \( {H}_{s}{H}_{s + 1} \) of \( P \) corresponds to an \( {H}_{s} - {H}_{s + 1} \) subpath of \( {V}_{i} \) that has no inner vertex on any path in \( \mathcal{H} \) . Together with (parts of) the \( i \) th segments of \( {H}_{2},\ldots ,{H}_{t - 1} \), these subpaths of \( {V}_{i} \) form an \( {H}^{j} - {H}^{j + 1} \) path \( {P}^{\prime } \) in \( G \) that has no inner vertices on any of the paths \( {H}^{1},\ldots ,{H}^{r} \) and meets no path from \( \mathcal{H} \) outside its \( i \) th segment. Replacing the ends of \( {P}^{\prime } \) on \( {H}^{j} \) and \( {H}^{j + 1} \) with \( {v}_{k}^{j} \) and \( {v}_{k}^{j + 1} \), respectively, we obtain our desired path \( {V}_{k}^{j} \) forming the \( j \) th (subdivided) edge of the \( k \) th ’vertical’ path of our grid. Since the paths \( {P}^{\prime } \) are disjoint for different \( i \) and different pairs \( \left( {j, k}\right) \) give rise to different \( i \), the paths \( {V}_{k}^{j} \) are disjoint except for possible common ends \( {v}_{k}^{j} \) . Moreover, they have no inner vertices on any of the paths \( {H}^{1},\ldots ,{H}^{r} \), because none of these \( {H}^{j} \) is an inner vertex of any of the paths \( P \subseteq {T}_{i} \) used in the construction of \( {V}_{k}^{j} \) . (3.3.1) Proof of Theorem 12.4.4. We are now ready to prove the following quantitative version of our theorem (which clearly implies it): Let \( r, m > 0 \) be integers, and let \( G \) be a graph of tree-width at least \( {r}^{4{m}^{2}\left( {r + 2}\right) } \) . Then \( G \) contains either the \( r \times r \) grid or \( {K}^{m} \) as a minor. Since \( {K}^{{r}^{2}} \) contains the \( r \times r \) grid as a subgraph we may assume that \( 2 \leq m \leq {r}^{2} \) . Put \( c \mathrel{\text{:=}} {r}^{4\left( {r + 2}\right) } \), and let \( k \mathrel{\text{:=}} {c}^{2\left( \begin{matrix} m \\ 2 \end{matrix}\right) } \) . Then \( c \geq {2}^{16} \) and hence \( {2m} + 3 \leq {c}^{m} \), so \( G \) has tree-width at least \[ {c}^{{m}^{2}} = {c}^{m}k \geq \left( {{2m} + 3}\right) k \geq \left( {m + 1}\right) \left( {{2k} - 1}\right) + k - 1, \] enough for Lemma 12.4.5 to ensure that \( G \) contains a \( k \) -mesh \( \left( {A, B}\right) \) <table><tr><td>\( \left( {A, B}\right) \)</td></tr><tr><td>\( T \)</td></tr><tr><td>\( X \)</td></tr><tr><td>\( {A}_{1},\ldots ,{A}_{m} \)</td></tr><tr><td>\( {\mathcal{P}}_{ij} \)</td></tr><tr><td>\( \sigma \)</td></tr><tr><td>\( {\ell }^{ * } \)</td></tr><tr><td>\( {H}_{ij}^{\ell } \)</td></tr><tr><td>\( {P}_{ij} \)</td></tr></table> of order \( \left( {m + 1}\right) \left( {{2k} - 1}\right) \) . Let \( T \subseteq A \) be a tree associated with the premesh \( \left( {A, B}\right) \) ; then \( X \mathrel{\text{:=}} V\left( {A \cap B}\right) \subseteq V\left( T\right) \) . By Lemma 12.4.6, \( T \) has \( \left| X\right| /\left( {{2k} - 1}\right) - 1 = m \) disjoint subtrees each containing at least \( k \) vertices of \( X \) ; let \( {A}_{1},\ldots ,{A}_{m} \) be the vertex sets of these trees. By definition of a \( k \) -mesh, \( B \) contains for all \( 1 \leq i < j \leq m \) a set \( {\mathcal{P}}_{ij} \) of \( k \) disjoint \( {A}_{i} - {A}_{j} \) paths that have no inner vertices in \( A \) . These sets \( {\mathcal{P}}_{ij} \) will shrink a little and be otherwise modified later in the proof, but they will always consist of ’many’ disjoint \( {A}_{i} - {A}_{j} \) paths. One option in our proof will be to find single paths \( {P}_{ij} \in {\mathcal{P}}_{ij} \) that are disjoint for different pairs \( {ij} \) and thus link up the sets \( {A}_{i} \) to form a \( {K}^{m} \) minor of \( G \) . If this fails, we shall instead exhibit two specific sets \( {\mathcal{P}}_{ij} \) and \( {\mathcal{P}}_{pq} \) such that many paths of \( {\mathcal{P}}_{ij} \) meet many paths of \( {\mathcal{P}}_{pq} \), forming an \( r \times r \) grid between them by Lemma 12.4.9. Let us impose a linear ordering on the index pairs \( {ij} \) by fixing an arbitrary bijection \( \sigma : \{ {ij} \mid 1 \leq i < j \leq m\} \rightarrow \left\{ {0,1,\ldots ,\left( \begin{matrix} m \\ 2 \end{matrix}\right) - 1}\right\} \) . For \( \ell = 0,1,\ldots \) in turn, we shall consider the pair \( {pq} \) with \( \sigma \left( {pq}\right) = \ell \) and choose an \( {A}_{p} - {A}_{q} \) path \( {P}_{pq} \) that is disjoint from all previously selected such paths, i.e. from the paths \( {P}_{st} \) with \( \sigma \left( {st}\right) < \ell \) . At the same time, we shall replace all the ’later’ sets \( {\mathcal{P}}_{ij} \) -or what has become of them-by smaller sets containing only paths

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

m} \) minor of \( G \) . If this fails, we shall instead exhibit two specific sets \( {\mathcal{P}}_{ij} \) and \( {\mathcal{P}}_{pq} \) such that many paths of \( {\mathcal{P}}_{ij} \) meet many paths of \( {\mathcal{P}}_{pq} \), forming an \( r \times r \) grid between them by Lemma 12.4.9. Let us impose a linear ordering on the index pairs \( {ij} \) by fixing an arbitrary bijection \( \sigma : \{ {ij} \mid 1 \leq i < j \leq m\} \rightarrow \left\{ {0,1,\ldots ,\left( \begin{matrix} m \\ 2 \end{matrix}\right) - 1}\right\} \) . For \( \ell = 0,1,\ldots \) in turn, we shall consider the pair \( {pq} \) with \( \sigma \left( {pq}\right) = \ell \) and choose an \( {A}_{p} - {A}_{q} \) path \( {P}_{pq} \) that is disjoint from all previously selected such paths, i.e. from the paths \( {P}_{st} \) with \( \sigma \left( {st}\right) < \ell \) . At the same time, we shall replace all the ’later’ sets \( {\mathcal{P}}_{ij} \) -or what has become of them-by smaller sets containing only paths that are disjoint from \( {P}_{pq} \) . Thus for each pair \( {ij} \), we shall define a sequence \( {\mathcal{P}}_{ij} = {\mathcal{P}}_{ij}^{0},{\mathcal{P}}_{ij}^{1},\ldots \) of smaller and smaller sets of paths, which eventually collapses to \( {\mathcal{P}}_{ij}^{\ell } = \left\{ {P}_{ij}\right\} \) when \( \ell \) has risen to \( \ell = \sigma \left( {ij}\right) \) . More formally, let \( {\ell }^{ * } \leq \left( \begin{matrix} m \\ 2 \end{matrix}\right) \) be the greatest integer such that, for all \( 0 \leq \ell < {\ell }^{ * } \) and all \( 1 \leq i < j \leq m \), there exist sets \( {\mathcal{P}}_{ij}^{\ell } \) satisfying the following five conditions: (i) \( {\mathcal{P}}_{ij}^{\ell } \) is a non-empty set of disjoint \( {A}_{i} - {A}_{j} \) paths in \( B \) that meet \( A \) only in their endpoints. Whenever a set \( {\mathcal{P}}_{ij}^{\ell } \) is defined, we shall write \( {H}_{ij}^{\ell } \mathrel{\text{:=}} \bigcup {\mathcal{P}}_{ij}^{\ell } \) for the union of its paths. (ii) If \( \sigma \left( {ij}\right) < \ell \) then \( {\mathcal{P}}_{ij}^{\ell } \) has exactly one element \( {P}_{ij} \), and \( {P}_{ij} \) does not meet any path belonging to a set \( {\mathcal{P}}_{st}^{\ell } \) with \( {ij} \neq {st} \) . (iii) If \( \sigma \left( {ij}\right) = \ell \), then \( \left| {\mathcal{P}}_{ij}^{\ell }\right| = k/{c}^{2\ell } \) . (iv) If \( \sigma \left( {ij}\right) > \ell \), then \( \left| {\mathcal{P}}_{ij}^{\ell }\right| = k/{c}^{2\ell + 1} \) . (v) If \( \ell = \sigma \left( {pq}\right) < \sigma \left( {ij}\right) \), then for every \( e \in E\left( {H}_{ij}^{\ell }\right) \smallsetminus E\left( {H}_{pq}^{\ell }\right) \) there are no \( k/{c}^{2\ell + 1} \) disjoint \( {A}_{i} - {A}_{j} \) paths in the graph \( \left( {{H}_{pq}^{\ell } \cup {H}_{ij}^{\ell }}\right) - e \) . Note that, by (iv), the paths considered in (v) do exist in \( {H}_{ij}^{\ell } \) . The purpose of (v) is to force those paths to reuse edges from \( {H}_{pq}^{{\ell }^{\prime }} \) whenever possible, using new edges \( e \notin {H}_{pq}^{\ell } \) only if necessary. Note further that since \( \sigma \left( {ij}\right) < \left( \begin{matrix} m \\ 2 \end{matrix}\right) \) by definition of \( \sigma \), conditions (iii) and (iv) give \( \left| {\mathcal{P}}_{ij}^{\ell }\right| \geq {c}^{2} \) whenever \( \sigma \left( {ij}\right) \geq \ell \) . Clearly if \( {\ell }^{ * } = \left( \begin{matrix} m \\ 2 \end{matrix}\right) \) then by (i) and (ii) we have a (subdivided) \( {K}^{m} \) minor with branch sets \( {A}_{1},\ldots ,{A}_{m} \) in \( G \) . Suppose then that \( {\ell }^{ * } < \left( \begin{matrix} m \\ 2 \end{matrix}\right) \) . Let us show that \( {\ell }^{ * } > 0 \) . Let \( {pq} \mathrel{\text{:=}} {\sigma }^{-1}\left( 0\right) \) and put \( {\mathcal{P}}_{pq}^{0} \mathrel{\text{:=}} {\mathcal{P}}_{pq} \) . To define \( {\mathcal{P}}_{ij}^{0} \) for \( \sigma \left( {ij}\right) > 0 \) put \( {H}_{ij} \mathrel{\text{:=}} \bigcup {\mathcal{P}}_{ij} \), let \( F \subseteq E\left( {H}_{ij}\right) \smallsetminus E\left( {H}_{pq}^{0}\right) \) be maximal such that \( \left( {{H}_{pq}^{0} \cup {H}_{ij}}\right) - F \) still contains \( k/c \) disjoint \( {A}_{i} - {A}_{j} \) paths, and let \( {\mathcal{P}}_{ij}^{0} \) be such a set of paths. Since the vertices from \( {A}_{p} \cup {A}_{q} \) have degree 1 in \( {H}_{pq}^{0} \cup {H}_{ij} \) unless they also lie in \( {A}_{i} \cup {A}_{j} \), these paths have no inner vertices in \( A \) . Our choices of \( {\mathcal{P}}_{ij}^{0} \) therefore satisfy (i)-(v) for \( \ell = 0 \) . Having shown that \( {\ell }^{ * } > 0 \), let us now consider \( \ell \mathrel{\text{:=}} {\ell }^{ * } - 1 \) . Thus, conditions (i)-(v) are satisfied for \( \ell \) but cannot be satisfied for \( \ell + 1 \) . Let \( {pq} \mathrel{\text{:=}} {\sigma }^{-1}\left( \ell \right) \) . If \( {\mathcal{P}}_{pq}^{\ell } \) contains a path \( P \) that avoids a set \( {\mathcal{Q}}_{ij} \) of some \( \left| {\mathcal{P}}_{ij}^{\ell }\right| /c \) of the paths in \( {\mathcal{P}}_{ij}^{\ell } \) for all \( {ij} \) with \( \sigma \left( {ij}\right) > \ell \), then we can define \( {\mathcal{P}}_{ij}^{\ell + 1} \) for all \( {ij} \) as before (with a contradiction). Indeed, let \( {st} \mathrel{\text{:=}} \) \( {\sigma }^{-1}\left( {\ell + 1}\right) \) and put \( {\mathcal{P}}_{st}^{\ell + 1} \mathrel{\text{:=}} {\mathcal{Q}}_{st} \) . For \( \sigma \left( {ij}\right) > \ell + 1 \) write \( {H}_{ij} \mathrel{\text{:=}} \bigcup {\mathcal{Q}}_{ij} \) , let \( F \subseteq E\left( {H}_{ij}\right) \smallsetminus E\left( {H}_{st}^{\ell + 1}\right) \) be maximal such that \( \left( {{H}_{st}^{\ell + 1} \cup {H}_{ij}}\right) - F \) still contains at least \( \left| {\mathcal{P}}_{ij}^{\ell }\right| /{c}^{2} \) disjoint \( {A}_{i} - {A}_{j} \) paths, and let \( {\mathcal{P}}_{ij}^{\ell + 1} \) be such a set of paths. Setting \( {\mathcal{P}}_{pq}^{\ell + 1} \mathrel{\text{:=}} \{ P\} \) and \( {\mathcal{P}}_{ij}^{\ell + 1} \mathrel{\text{:=}} {\mathcal{P}}_{ij}^{\ell } = \left\{ {P}_{ij}^{{\ell }^{\prime }}\right\} \) for \( \sigma \left( {ij}\right) < \ell \) then gives us a family of sets \( {\mathcal{P}}_{ij}^{\ell + 1} \) that contradicts the maximality of \( {\ell }^{ * } \) . Thus for every path \( P \in {\mathcal{P}}_{pq}^{\ell } \) there exists a pair \( {ij} \) with \( \sigma \left( {ij}\right) > \ell \) such that \( P \) avoids fewer than \( \left| {\mathcal{P}}_{ij}^{\ell }\right| /c \) of the paths in \( {\mathcal{P}}_{ij}^{\ell } \) . For some \( \left\lceil {\left| {\mathcal{P}}_{pq}^{\ell }\right| /\left( \begin{matrix} m \\ 2 \end{matrix}\right) }\right\rceil \) of these \( P \) that pair \( {ij} \) will be the same; let \( \mathcal{P} \) denote the set of those \( P \), and keep \( {ij} \) fixed from now on. Note that \( \left| \mathcal{P}\right| \geq \left| {\mathcal{P}}_{pq}^{\ell }\right| /\left( \begin{matrix} m \\ 2 \end{matrix}\right) = \) \( c\left| {\mathcal{P}}_{ij}^{\ell }\right| /\left( \begin{matrix} m \\ 2 \end{matrix}\right) \) by (iii) and (iv). Let us use Lemma 12.4.7 to find sets \( \mathcal{V} \subseteq \mathcal{P} \subseteq {\mathcal{P}}_{pq}^{\ell } \) and \( \mathcal{H} \subseteq {\mathcal{P}}_{ij}^{\ell } \) such that \[ \left| \mathcal{V}\right| \geq \frac{1}{2}\left| \mathcal{P}\right| \;\left( { \geq \frac{c}{{m}^{2}}\left| {\mathcal{P}}_{ij}^{\ell }\right| }\right) \] \[ \left| \mathcal{H}\right| = {r}^{2} \] and every path in \( \mathcal{V} \) meets every path in \( \mathcal{H} \) . We have to check that the bipartite graph with vertex sets \( \mathcal{P} \) and \( {\mathcal{P}}_{ij}^{\ell } \) in which \( P \in \mathcal{P} \) is adjacent to \( Q \in {\mathcal{P}}_{ij}^{\ell } \) whenever \( P \cap Q = \varnothing \) does not have too many edges. Since every \( P \in \mathcal{P} \) has fewer than \( \left| {\mathcal{P}}_{ij}^{\ell }\right| /c \) neighbours (by definition of \( \mathcal{P} \) ), this graph indeed has at most \[ \left| \mathcal{P}\right| \left| {\mathcal{P}}_{ij}^{\ell }\right| /c \leq \left| \mathcal{P}\right| \left| {\mathcal{P}}_{ij}^{\ell }\right| /6{r}^{2} \] \[ \leq \lfloor \left| \mathcal{P}\right| /2\rfloor \left| {\mathcal{P}}_{ij}^{\ell }\right| /2{r}^{2} \] \[ \leq \lfloor \left| \mathcal{P}\right| /2\rfloor \left( {\left| {\mathcal{P}}_{ij}^{\ell }\right| /{r}^{2} - 1}\right) \] \[ = \left( {\left| \mathcal{P}\right| -\lceil \left| \mathcal{P}\right| /2\rceil }\right) \left( {\left| {\mathcal{P}}_{ij}^{\ell }\right| - {r}^{2}}\right) /{r}^{2} \] edges, as required. Hence, \( \mathcal{V} \) and \( \mathcal{H} \) exist as claimed. Although all the (’vertical’) paths in \( \mathcal{V} \) meet all the (’horizontal’) paths in \( \mathcal{H} \), these paths do not necessarily intersect in such an orderly way as required for Lemma 12.4.9. In order to divide the paths from \( \mathcal{H} \) into segments, and to select paths from \( \mathcal{V} \) meeting them only in the appropriate segments, we shall first pick a path \( Q \in \mathcal{H} \) to serve as a yardstick: we shall divide \( Q \) into segments each meeting lots of paths from \( \mathcal{V} \), select a ’non-crossing’ subset \( {V}_{1},\ldots ,{V}_{d} \) of these vertical paths, one from each segment (which is the most delicate task; we shall need condition (v) from the definition of the sets \( {\mathcal{P}}_{ij}^{\ell } \) here), and finally divide the other horizontal paths into the 'induced' segments, accommodating one \( {V}_{n} \) each. So let us pick a path \( Q \in \mathcal{H} \), and put \[ d \mathrel{\text{:=}} \lfloor \sqrt{c}/m\rfloor = \left\lfloor {{r}^{{2r} + 4}/m}\right\rfloor \geq {r}^{{2r} + 2}. \] Note that \( \left| \mathcal{V}\right| \geq \left( {c/{m}^{2}}\right) \left| {\mathcal{P}}_{ij}^{\ell }\right| \geq {d}^{2}\left| {\mathcal{P}}_{ij}^{\ell }\right| \) . For \( n = 1,2,\ldots, d - 1 \) let \( {e}_{n} \) be the first edge of \( Q \) (on its way from

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

only in the appropriate segments, we shall first pick a path \( Q \in \mathcal{H} \) to serve as a yardstick: we shall divide \( Q \) into segments each meeting lots of paths from \( \mathcal{V} \), select a ’non-crossing’ subset \( {V}_{1},\ldots ,{V}_{d} \) of these vertical paths, one from each segment (which is the most delicate task; we shall need condition (v) from the definition of the sets \( {\mathcal{P}}_{ij}^{\ell } \) here), and finally divide the other horizontal paths into the 'induced' segments, accommodating one \( {V}_{n} \) each. So let us pick a path \( Q \in \mathcal{H} \), and put \[ d \mathrel{\text{:=}} \lfloor \sqrt{c}/m\rfloor = \left\lfloor {{r}^{{2r} + 4}/m}\right\rfloor \geq {r}^{{2r} + 2}. \] Note that \( \left| \mathcal{V}\right| \geq \left( {c/{m}^{2}}\right) \left| {\mathcal{P}}_{ij}^{\ell }\right| \geq {d}^{2}\left| {\mathcal{P}}_{ij}^{\ell }\right| \) . For \( n = 1,2,\ldots, d - 1 \) let \( {e}_{n} \) be the first edge of \( Q \) (on its way from \( {e}_{n} \) \( \left. {{A}_{i}\text{to}{A}_{j}}\right) \) such that the initial component \( {Q}_{n} \) of \( Q - {e}_{n} \) meets at least \( {Q}_{n} \) \( {nd}\left| {\mathcal{P}}_{ij}^{\ell }\right| \) different paths from \( \mathcal{V} \), and such that \( {e}_{n} \) is not an edge of \( {H}_{pq}^{\ell } \) . As any two vertices of \( Q \) that lie on different paths from \( \mathcal{V} \) are separated in \( Q \) by an edge not in \( {H}_{pq}^{\ell } \), each of these \( {Q}_{n} \) meets exactly \( {nd}\left| {\mathcal{P}}_{ij}^{\ell }\right| \) paths from \( \mathcal{V} \) . Put \( {Q}_{0} \mathrel{\text{:=}} \varnothing \) and \( {Q}_{d} \mathrel{\text{:=}} Q \) . Since \( \left| \mathcal{V}\right| \geq {d}^{2}\left| {\mathcal{P}}_{ij}^{\ell }\right| \), we have thus divided \( Q \) into \( d \) consecutive disjoint segments \( {Q}_{n}^{\prime } \mathrel{\text{:=}} {Q}_{n} - {Q}_{n - 1} \) \( \left( {n = 1,\ldots, d}\right) \) each meeting at least \( d\left| {\mathcal{P}}_{ij}^{\ell }\right| \) paths from \( \mathcal{V} \) . \[ {Q}_{1}^{\prime },\ldots ,{Q}_{d}^{\prime } \] For each \( n = 1,\ldots, d - 1 \), Menger’s theorem (3.3.1) and conditions (iv) and (v) imply that \( {H}_{pq}^{\ell } \cup {H}_{ij}^{\ell } \) has a set \( {S}_{n} \) of \( \left| {\mathcal{P}}_{ij}^{\ell }\right| - 1 \) vertices such that \( \left( {{H}_{pq}^{\ell } \cup {H}_{ij}^{\ell }}\right) - {e}_{n} - {S}_{n} \) contains no path from \( {A}_{i} \) to \( {A}_{j} \) . Let \( S \) denote the union of all these sets \( {S}_{n} \) . Then \( \left| S\right| < d\left| {\mathcal{P}}_{ij}^{\ell }\right| \), so each \( {Q}_{n}^{\prime } \) meets at least one path \( {V}_{n} \in \mathcal{V} \) that avoids \( S \) (Fig. 12.4.5).  Fig. 12.4.5. \( {V}_{n} \) meets every horizontal path but avoids \( S \) Clearly, each \( {S}_{n} \) consists of a choice of exactly one vertex \( x \) from every path \( P \in {\mathcal{P}}_{ij}^{\ell } \smallsetminus \{ Q\} \) . Denote the initial component of \( P - x \) by \( {P}_{n} \) , \( {P}_{1}^{\prime },\ldots ,{P}_{d}^{\prime } \) put \( {P}_{0} \mathrel{\text{:=}} \varnothing \) and \( {P}_{d} \mathrel{\text{:=}} P \), and let \( {P}_{n}^{\prime } \mathrel{\text{:=}} {P}_{n} - {P}_{n - 1} \) for \( n = 1,\ldots, d \) . The separation properties of the sets \( {S}_{n} \) now imply that \( {V}_{n} \cap P \subseteq {P}_{n}^{\prime } \) for \( n = 1,\ldots, d \) (and hence in particular that \( {P}_{n}^{\prime } \neq \varnothing \), i.e. that \( {P}_{n - 1} \subset {P}_{n} \) ). Indeed \( {V}_{n} \) cannot meet \( {P}_{n - 1} \), because \( {P}_{n - 1} \cup {V}_{n} \cup \left( {Q - {Q}_{n - 1}}\right) \) would then contain an \( {A}_{i} - {A}_{j} \) path in \( \left( {{H}_{pq}^{\ell } \cup {H}_{ij}^{\ell }}\right) - {e}_{n - 1} - {S}_{n - 1} \), and likewise (consider \( {S}_{n} \) ) \( {V}_{n} \) cannot meet \( P - {P}_{n} \) . Thus for all \( n = 1,\ldots, d \), the path \( {V}_{n} \) meets every path \( P \in \mathcal{H} \smallsetminus \{ Q\} \) precisely in its \( n \) th segment \( {P}_{n}^{\prime } \) . Applying Lemma 12.4.9 to the path systems \( \mathcal{H} \smallsetminus \{ Q\} \) and \( \left\{ {{V}_{1},\ldots ,{V}_{d}}\right\} \) now yields the desired grid minor. Theorem 12.4.3 has an interesting application. Recall that a class \( \mathcal{H} \) of graphs has the Erdős-Pósa property if the number of vertices in a graph needed to cover all its subgraphs in \( \mathcal{H} \) is bounded by a function of its maximum number of disjoint subgraphs in \( \mathcal{H} \) . Now let \( H \) be a fixed connected graph, and consider the class \( \mathcal{H} = {MH} \) of graphs that contract to a copy of \( H \) . (Thus, \( G \) has a subgraph in \( \mathcal{H} \) if and only if \( H \preccurlyeq G \) .) ## Corollary 12.4.10. If \( H \) is planar, then \( \mathcal{H} = {MH} \) has the Erdős-Pósa property. Proof. We have to find a function \( f : \mathbb{N} \rightarrow \mathbb{N} \) such that, given \( k \in \mathbb{N} \) and a graph \( G \), either \( G \) has \( k \) disjoint subgraphs in \( {MH} \) or there is a set \( U \) of at most \( f\left( k\right) \) vertices in \( G \) such that \( G - U \) has no subgraph in \( {MH} \) , i.e. \( H \npreceq G - U \) . By Theorem 12.4.3, there exists for every \( k \geq 1 \) an integer \( {w}_{k} \) such that every graph of tree-width at least \( {w}_{k} \) contains the disjoint union of \( k \) copies of \( H \) (which is again planar) as a minor. Define \[ f\left( k\right) \mathrel{\text{:=}} {2f}\left( {k - 1}\right) + {w}_{k} \] inductively, starting with \( f\left( 0\right) = f\left( 1\right) = 0 \) . To verify that \( f \) does what it should, we apply induction on \( k \) . For \( k \leq 1 \) there is nothing to show. Now let \( k \) and \( G \) be given for the induction step. If \( \operatorname{tw}\left( G\right) \geq {w}_{k} \), we are home by definition of \( {w}_{k} \) . So assume that \( \operatorname{tw}\left( G\right) < {w}_{k} \), and let \( \left( {T,{\left( {V}_{t}\right) }_{t \in T}}\right) \) be a tree-decomposition of \( G \) of width \( < {w}_{k} \) . Let us direct the edges \( {t}_{1}{t}_{2} \) of the tree \( T \) as follows. Let \( {T}_{1},{T}_{2} \) be the components of \( T - {t}_{1}{t}_{2} \) containing \( {t}_{1} \) and \( {t}_{2} \) , respectively, and put \[ {G}_{1} \mathrel{\text{:=}} G\left\lbrack {\mathop{\bigcup }\limits_{{t \in {T}_{1}}}\left( {{V}_{t} \smallsetminus {V}_{{t}_{2}}}\right) }\right\rbrack \;\text{ and }\;{G}_{2} \mathrel{\text{:=}} G\left\lbrack {\mathop{\bigcup }\limits_{{t \in {T}_{2}}}\left( {{V}_{t} \smallsetminus {V}_{{t}_{1}}}\right) }\right\rbrack \] We direct the edge \( {t}_{1}{t}_{2} \) towards \( {G}_{i} \) if \( H \preccurlyeq {G}_{i} \), thereby giving \( {t}_{1}{t}_{2} \) either one or both or neither direction. If every edge of \( T \) receives at most one direction, we follow these to a node \( t \in T \) such that no edge at \( t \) in \( T \) is directed away from \( t \) . As \( H \) is connected, this implies by Lemma 12.3.1 that \( {V}_{t} \) meets every \( {MH} \) in \( G \) . This completes the proof with \( U = {V}_{t} \), since \( \left| {V}_{t}\right| \leq {w}_{k} \leq f\left( k\right) \) by the choice of our tree-decomposition. Suppose now that \( T \) has an edge \( {t}_{1}{t}_{2} \) that received both directions. For each \( i = 1,2 \) let us ask if we can cover all the \( {MH} \) subgraphs of \( {G}_{i} \) by at most \( f\left( {k - 1}\right) \) vertices. If we can, for both \( i \), then by Lemma 12.3.1 the two covers combine with \( {V}_{{t}_{1}} \cap {V}_{{t}_{2}} \) to the desired cover \( U \) for \( G \) . Suppose now that \( {G}_{1} \) has no such cover. Then, by the induction hypothesis, \( {G}_{1} \) has \( k - 1 \) disjoint \( {MH} \) subgraphs. Since \( {t}_{1}{t}_{2} \) was also directed towards \( {t}_{2} \), there is another such subgraph in \( {G}_{2} \) . This gives the desired total of \( k \) disjoint \( {MH} \) subgraphs in \( G \) . Note that Corollary 12.4.10 contains the Erdős-Pósa theorem 2.3.2 as the special case of \( H = {K}^{3} \) . It is best possible in that if \( H \) is nonplanar, then \( {MH} \) does not have the Erdős-Pósa property (Exercise 39). We conclude this section with statements of the structure theorems for the graphs not containing a given complete graph as a minor. These are far more difficult to prove than any of the results we have seen so far, and they are not even that easy to state. But it's worth an effort: the statement of the excluded- \( {K}^{n} \) theorem is interesting, it is central to the proof of the graph minor theorem, and it can be applied elsewhere. The torsos of a tree-decomposition \( \left( {T,{\left( {V}_{t}\right) }_{t \in T}}\right) \) of a graph \( G \) are torsos the graphs \( {H}_{t}\left( {t \in T}\right) \) obtained from \( G\left\lbrack {V}_{t}\right\rbrack \) by adding all the edges \( {xy} \) such that \( x, y \in {V}_{t} \cap {V}_{{t}^{\prime }} \) for some neighbour \( {t}^{\prime } \) of \( t \) in \( T \) . (Thus, if a tree-decomposition happens to be simplicial, its torsos are just its parts.) --- linear decomposition --- A linear decomposition of \( G \) is a family \( {\left( {V}_{i}\right) }_{i \in I} \) of vertex sets indexed by some linear order \( I \) such that \( \mathop{\bigcup }\limits_{{i \in I}}{V}_{i} = V\left( G\right) \), every edge of \( G \) has both its ends in some \( {V}_{i} \), and \( {V}_{i} \cap {V}_{k} \subseteq {V}_{j} \) whenever \( i < j < k \) . When \( G \) is finite, this is just a tree-decomposition whose decomposition tree is a path, and usually called a path-decomposition. If each \( {V}_{i} \) contains at most \( k \) vertices and \( k \) is minimal with this property, then \( {\left( {V}_{i}\right) }_{i \in I} \) has width \( k - 1 \) . Let \( {S}^{\prime }

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G\left\lbrack {V}_{t}\right\rbrack \) by adding all the edges \( {xy} \) such that \( x, y \in {V}_{t} \cap {V}_{{t}^{\prime }} \) for some neighbour \( {t}^{\prime } \) of \( t \) in \( T \) . (Thus, if a tree-decomposition happens to be simplicial, its torsos are just its parts.) --- linear decomposition --- A linear decomposition of \( G \) is a family \( {\left( {V}_{i}\right) }_{i \in I} \) of vertex sets indexed by some linear order \( I \) such that \( \mathop{\bigcup }\limits_{{i \in I}}{V}_{i} = V\left( G\right) \), every edge of \( G \) has both its ends in some \( {V}_{i} \), and \( {V}_{i} \cap {V}_{k} \subseteq {V}_{j} \) whenever \( i < j < k \) . When \( G \) is finite, this is just a tree-decomposition whose decomposition tree is a path, and usually called a path-decomposition. If each \( {V}_{i} \) contains at most \( k \) vertices and \( k \) is minimal with this property, then \( {\left( {V}_{i}\right) }_{i \in I} \) has width \( k - 1 \) . Let \( {S}^{\prime } \) be a subspace of a surface \( {}^{6}S \) obtained by removing the interiors of finitely many disjoint closed discs, with boundary circles \( {C}_{1},\ldots ,{C}_{k} \) say. This space is determined up to homeomorphism by \( S \) --- \( {C}_{1},\ldots ,{C}_{k} \) \( S - k \) --- and the number \( k \), and we denote it by \( S - k \) . Each \( {C}_{i} \) is the image of a continuous map \( {f}_{i} : \left\lbrack {0,1}\right\rbrack \rightarrow {S}^{\prime } \) that is injective except for \( {f}_{i}\left( 0\right) = {f}_{i}\left( 1\right) \) . We call \( {C}_{1},\ldots ,{C}_{k} \) the cuffs of \( {S}^{\prime } \) and the points \( {f}_{1}\left( 0\right) ,\ldots ,{f}_{k}\left( 0\right) \) their cuffs roots. The other points of each \( {C}_{i} \) are linearly ordered by \( {f}_{i} \) as images of \( \left( {0,1}\right) \) ; when we use cuffs as index sets for linear decompositions below, we shall be referring to these linear orders. --- \( {}^{6} \) A compact connected 2-manifold without boundary; see Appendix B. --- --- \( k \) -near embedding --- Let \( H \) be a graph, \( S \) a surface, and \( k \in \mathbb{N} \) . We say that \( H \) is \( k \) -nearly embeddable in \( S \) if \( H \) has a set \( X \) of at most \( k \) vertices such that \( H - X \) can be written as \( {H}_{0} \cup {H}_{1} \cup \ldots \cup {H}_{k} \) so that (N1) there exists an embedding \( \sigma : {H}_{0} \hookrightarrow S - k \) that maps only vertices to cuffs and no vertex to the root of a cuff; (N2) the graphs \( {H}_{1},\ldots ,{H}_{k} \) are pairwise disjoint (and may be empty), and \( {H}_{0} \cap {H}_{i} = {\sigma }^{-1}\left( {C}_{i}\right) \) for each \( i \) ; (N3) every \( {H}_{i} \) with \( i \geq 1 \) has a linear decomposition \( {\left( {V}_{z}^{i}\right) }_{z \in {C}_{i} \cap \sigma \left( {H}_{0}\right) } \) of width at most \( k \) such that \( z \in {V}_{z}^{i} \) for all \( z \) . Here, then, is the structure theorem for the graphs without a \( {K}^{n} \) minor: Theorem 12.4.11. (Robertson & Seymour 2003) For every \( n \in \mathbb{N} \) there exists a \( k \in \mathbb{N} \) such that every graph \( G \) not containing \( {K}^{n} \) as a minor has a tree-decomposition whose torsos are \( k \) -nearly embeddable in a surface in which \( {K}^{n} \) is not embeddable. Note that there are only finitely many surfaces in which \( {K}^{n} \) is not embeddable. The set of those surfaces in the statement of Theorem 12.4.11 could therefore be replaced by just two surfaces: the orientable and the non-orientable surface of maximum genus in this set. Note also that the separators \( {V}_{t} \cap {V}_{{t}^{\prime }} \) in the tree-decomposition of \( G \) (for edges \( t{t}^{\prime } \) of the decomposition tree) have bounded size, e.g. at most \( {2k} + n \) , because they induce complete subgraphs in the torsos and these are \( k \) - nearly embeddable in one of those two surfaces. We remark that Theorem 12.4.11 has only a qualitative converse: graphs that admit a decomposition as described can clearly have a \( {K}^{n} \) minor, but there exists an integer \( r \) depending only on \( n \) such that none of them has a \( {K}^{r} \) minor. Theorem 12.4.11, as stated above, is true also for infinite graphs (Diestel & Thomas 1999). There are also structure theorems for excluding infinite minors, and we state two of these. First, the structure theorem for excluding \( {K}^{{\aleph }_{0}} \) . Call a graph \( H \) --- nearly planar --- nearly planar if \( H \) has a finite set \( X \) of vertices such that \( H - X \) can be written as \( {H}_{0} \cup {H}_{1} \) so that \( \left( {\mathrm{N}1 - 2}\right) \) hold with \( S = {S}^{2} \) (the sphere) and \( k = 1 \), while (N3) holds with \( k = \left| X\right| \) . (In other words, deleting a bounded number of vertices makes \( H \) planar except for a subgraph of bounded linear width sewn on to the unique cuff of \( {S}^{2} - 1 \) .) A tree-adhesion decomposition \( \left( {T,{\left( {V}_{t}\right) }_{t \in T}}\right) \) of a graph \( G \) has finite adhesion if for every edge \( t{t}^{\prime } \in T \) the set \( {V}_{t} \cap {V}_{{t}^{\prime }} \) is finite and for every infinite path \( {t}_{1}{t}_{2}\ldots \) in \( T \) the value of \( \lim \mathop{\inf }\limits_{{i \rightarrow \infty }}\left| {{V}_{{t}_{i}} \cap {V}_{{t}_{i + 1}}}\right| \) is finite. Unlike its counterpart for \( {K}^{n} \), the excluded- \( {K}^{{\aleph }_{0}} \) structure theorem has a direct converse. It thus characterizes the graphs without a \( {K}^{{\aleph }_{0}} \) minor, as follows: Theorem 12.4.12. (Diestel, Robertson, Seymour & Thomas 1995-99) A graph \( G \) has no \( {K}^{{\aleph }_{0}} \) minor if and only if \( G \) has a tree-decomposition of finite adhesion whose torsos are nearly planar. Finally, a structure theorem for excluding \( {K}^{{\aleph }_{0}} \) as a topological mi- --- finite tree-width --- nor. Let us say that \( G \) has finite tree-width if \( G \) admits a tree-decomposition \( \left( {T,{\left( {V}_{t}\right) }_{t \in T}}\right) \) into finite parts such that for every infinite path \( {t}_{1}{t}_{2}\ldots \) in \( T \) the set \( \mathop{\bigcup }\limits_{{j \geq 1}}\mathop{\bigcap }\limits_{{i \geq j}}{V}_{{t}_{i}} \) is finite. Theorem 12.4.13. (Diestel, Robertson, Seymour & Thomas 1992-94) The following assertions are equivalent for connected graphs \( G \) : (i) \( G \) does not contain \( {K}^{{\aleph }_{0}} \) as a topological minor; (ii) \( G \) has finite tree-width; (iii) \( G \) has a normal spanning tree \( T \) such that for every ray \( R \) in \( T \) there are only finitely many vertices \( v \) that can be linked to \( R \) by infinitely many paths meeting pairwise only in \( v \) . ## 12.5 The graph minor theorem Graph properties that are closed under taking minors occur frequently in graph theory. Among the most natural examples are the properties of being embeddable in some fixed surface, such as planarity. By Kuratowski's theorem, planarity can be expressed by forbidding the minors \( {K}^{5} \) and \( {K}_{3,3} \) . This is a good characterization of planarity in the following sense. Suppose we wish to persuade someone that a certain graph is planar: this is easy (at least intuitively) if we can produce a drawing of the graph. But how do we persuade someone that a graph is non-planar? By Kuratowski's theorem, there is also an easy way to do that: we just have to exhibit an \( M{K}^{5} \) or \( M{K}_{3,3} \) in our graph, as an easily checked 'certificate' for non-planarity. Our simple Proposition 12.4.2 is another example of a good characterization: if a graph has tree width \( < 3 \), we can prove this by exhibiting a suitable tree-decomposition; if not, we can produce an \( M{K}^{4} \) as evidence. Theorems that characterize a property \( \mathcal{P} \) by a set of forbidden minors are doubtless among the most attractive results in graph theory. As we saw in Proposition 12.4.1, such a characterization exists whenever \( \mathcal{P} \) (12.4.1) is minor-closed: then \( \mathcal{P} = {\operatorname{Forb}}_{ \preccurlyeq }\left( \overline{\mathcal{P}}\right) \), where \( \overline{\mathcal{P}} \) is the complement of \( \mathcal{P} \) . However, one naturally seeks to make the set of forbidden minors as small as possible. And there is indeed a unique smallest such set: the set \[ {\mathcal{K}}_{\mathcal{P}} \mathrel{\text{:=}} \{ H \mid H\text{ is } \preccurlyeq \text{-minimal in }\overline{\mathcal{P}}\} \] --- Kuratowski set \( {\mathcal{K}}_{\mathcal{P}} \) --- satisfies \( \mathcal{P} = {\operatorname{Forb}}_{ \preccurlyeq }\left( {\mathcal{K}}_{\mathcal{P}}\right) \) and is contained in every other set \( \mathcal{H} \) such that \( \mathcal{P} = {\operatorname{Forb}}_{ \preccurlyeq }\left( \mathcal{H}\right) \) . We call \( {\mathcal{K}}_{\mathcal{P}} \) the Kuratowski set for \( \mathcal{P} \) . Clearly, the elements of \( {\mathcal{K}}_{\mathcal{P}} \) are incomparable under the minor relation \( \preccurlyeq \) . Now the graph minor theorem of Robertson &Seymour says that any set of \( \preccurlyeq \) -incomparable graphs must be finite: --- graph minor theorem --- Theorem 12.5.1. (Robertson & Seymour 1986-2004) The finite graphs are well-quasi-ordered by the minor relation \( \preccurlyeq \) . We shall give a sketch of the proof of the graph minor theorem at the end of this section. Corollary 12.5.2. The Kuratowski set for any minor-closed graph property is finite. As a special case of Corollary 12.5.2 we have, at least in principle, a Kuratowski-type theorem for every surface \( S \) : the property \( \mathcal{P}\left( S\right) \) of embeddability in \( S \) is characterized by the finite set \( {\mathcal{K}}_{\mathcal{P}\left( S\right) } \) of forbidden minors. Corollary 12.5.3. For every surface \( S \) there exists a finite set of graphs \( {H}_{1},\ldots ,{H}_{n} \) such that a graph is embeddabl

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{K}}_{\mathcal{P}} \) are incomparable under the minor relation \( \preccurlyeq \) . Now the graph minor theorem of Robertson &Seymour says that any set of \( \preccurlyeq \) -incomparable graphs must be finite: --- graph minor theorem --- Theorem 12.5.1. (Robertson & Seymour 1986-2004) The finite graphs are well-quasi-ordered by the minor relation \( \preccurlyeq \) . We shall give a sketch of the proof of the graph minor theorem at the end of this section. Corollary 12.5.2. The Kuratowski set for any minor-closed graph property is finite. As a special case of Corollary 12.5.2 we have, at least in principle, a Kuratowski-type theorem for every surface \( S \) : the property \( \mathcal{P}\left( S\right) \) of embeddability in \( S \) is characterized by the finite set \( {\mathcal{K}}_{\mathcal{P}\left( S\right) } \) of forbidden minors. Corollary 12.5.3. For every surface \( S \) there exists a finite set of graphs \( {H}_{1},\ldots ,{H}_{n} \) such that a graph is embeddable in \( S \) if and only if it contains none of \( {H}_{1},\ldots ,{H}_{n} \) as a minor. The proof of Corollary 12.5.3 does not need the full strength of the minor theorem. We shall give a direct proof, which runs as follows. The main step is to prove that the graphs in \( {\mathcal{K}}_{\mathcal{P}\left( S\right) } \) do not contain arbitrarily large grids as minors (Lemma 12.5.4). Then their tree-width is bounded (Theorem 12.4.4), so \( {\mathcal{K}}_{\mathcal{P}\left( S\right) } \) is well-quasi-ordered (Theorem 12.3.7) and therefore finite. The proof of Lemma 12.5.4 gives a good impression of the interplay between graph minors and surface topology, which-by way of Theorem 12.4.11, which we could not prove here is also one of the key ingredients of the proof of the graph minor theorem. Appendix B summarizes the necessary background on surfaces, including a lemma. For convenience (cf. Proposition 1.7.2 (ii)), we shall work with hexagonal rather than square grids. \( {H}^{r} \) Denote by \( {H}^{r} \) the plane hexagonal grid whose dual has radius \( r \) (Figure 12.5.1). The face corresponding to the central vertex of its dual faces is its central face. (Generally, when we speak of the faces of \( {H}^{r} \), we mean its hexagonal faces, not its outer face.) A subgrid \( {H}^{k} \) of \( {H}^{r} \) is --- canonical \( {S}_{1},\ldots ,{S}_{r} \) ring \( {R}_{k} \) --- canonical if their central faces coincide. We write \( {S}_{k} \) for the perimeter cycle of the canonical subgrid \( {H}^{k} \) in \( {H}^{r} \) ; for example, \( {S}_{1} \) is the hexagon bounding the central face of \( {H}^{r} \) . The ring \( {R}_{k} \) is the subgraph of \( {H}^{r} \) formed by \( {S}_{k} \) and \( {S}_{k + 1} \) and the edges between them.  Fig. 12.5.1. The hexagonal grid \( {H}^{6} \) with central face \( f \) and rings \( {R}_{2} \) and \( {R}_{5} \) Lemma 12.5.4. For every surface \( S \) there exists an integer \( r \) such that no graph that is minimal with the property of not being embeddable in \( S \) contains \( {H}^{r} \) as a topological minor. (4.1.2) Proof. Let \( G \) be a graph that cannot be embedded in \( S \) and is minimal (4.2.2) (4.3.2) with this property. Our proof will run roughly as follows. Since \( G \) (App. B) is minimally not embeddable in \( S \), we can embed it in an only slightly larger surface \( {S}^{\prime } \) . If \( G \) contains a very large \( {H}^{r} \) grid, then by Lemma B. 6 some large \( {H}^{m} \) subgrid will be flat in \( {S}^{\prime } \), that is, the union of its faces in \( {S}^{\prime } \) will be a disc \( {D}^{\prime } \) . We then pick an edge \( e \) from the middle of this \( {H}^{m} \) grid and embed \( G - e \) in \( S \) . Again by Lemma B. 6, one of the rings of our \( {H}^{m} \) will be flat in \( S \) . In this ring we can embed the (planar) subgraph of \( G \) which our first embedding had placed in \( {D}^{\prime } \) ; note that this subgraph contains the edge \( e \) . The rest of \( G \) can then be embedded in \( S \) outside this ring much as before, yielding an embedding of all of \( G \) in \( S \) (a contradiction). More formally, let \( \varepsilon \mathrel{\text{:=}} \varepsilon \left( S\right) \) denote the Euler genus of \( S \) . Let \( r \) be large enough that \( {H}^{r} \) contains \( \varepsilon + 3 \) disjoint copies of \( {H}^{m + 1} \), where \( m \mathrel{\text{:=}} {3\varepsilon } + 4 \) . We show that \( G \) has no \( T{H}^{r} \) subgraph. \( r, m \) Let \( {e}^{\prime } = {u}^{\prime }{v}^{\prime } \) be any edge of \( G \), and choose an embedding \( {\sigma }^{\prime } \) of \( G - {e}^{\prime } \) in \( S \) . Choose a face with \( {u}^{\prime } \) on its boundary, and another with \( {v}^{\prime } \) on its boundary. Cut a disc out of each face and add a handle between the two holes, to obtain a surface \( {S}^{\prime } \) of Euler genus \( \varepsilon + 2 \) (Lemma B.3). Embedding \( {e}^{\prime } \) along this handle, extend \( {\sigma }^{\prime } \) to an embedding of \( G \) in \( {S}^{\prime } \) . \( {\sigma }^{\prime } : G \hookrightarrow {S}^{\prime } \) Suppose \( G \) has a subgraph \( H = T{H}^{r} \) . Let \( f : {H}^{r} \rightarrow H \) map the vertices of \( {H}^{r} \) to the corresponding branch vertices of \( H \), and its edges to the corresponding paths in \( H \) between those vertices. Let us show that \( {H}^{r} \) has a subgrid \( {H}^{m} \) (not necessarily canonical) whose hexagonal face boundaries correspond (by \( {\sigma }^{\prime } \circ f \) ) to circles in \( {S}^{\prime } \) that bound disjoint open discs there. By the choice of \( r \), we can find \( \varepsilon + 3 \) disjoint copies of \( {H}^{m + 1} \) in \( {H}^{r} \) . The canonical subgrids \( {H}^{m} \) of these \( {H}^{m + 1} \) are not only disjoint, but sufficiently spaced out in \( {H}^{r} \) that their deletion leaves a tree \( T \subseteq {H}^{r} \) that sends an edge to each of them (Figure 12.5.2). If each of these \( {H}^{m} \) has a face whose boundary maps to a circle in \( {S}^{\prime } \) not bounding a disc there, and \( \mathcal{C} \) denotes the set of those \( \varepsilon + 3 \) circles, then \( {S}^{\prime } \smallsetminus \bigcup \mathcal{C} \) has a component \( {D}_{0} \) whose closure meets every circle in \( \mathcal{C} \) : the component containing \( \left( {{\sigma }^{\prime } \circ f}\right) \left( T\right) \) . As \( \varepsilon \left( {S}^{\prime }\right) = \varepsilon + 2 \), this contradicts Lemma B.6. Hence for one of our copies of \( {H}^{m} \) in \( {H}^{r} \), every hexagon of \( {H}^{m} \) bounds an open disc in \( {S}^{\prime } \) . If these discs are not disjoint, then one of them, \( D \) say, meets the boundary of another such disc. But since the frontier \( C \) of \( D \) separates \( D \) in \( {S}^{\prime } \) from the rest of \( {S}^{\prime } \), and \( {\sigma }^{\prime }\left( H\right) \smallsetminus C \) is connected, this means that the closure of \( D \) contains the entire graph \( {\sigma }^{\prime }\left( H\right) \) . Contracting \( \left( {{\sigma }^{\prime } \circ f}\right) \left( {S}_{r}\right) \) in \( {\sigma }^{\prime }\left( H\right) \) now yields a 3-connected graph embedded in a disc. By Theorem 4.3.2, its faces correspond to those of \( {H}^{r}/{S}_{r} \) in the plane, i.e. are disjoint discs. Thus, \( {H}^{m} \) exists as claimed. From now on, we shall work with this fixed \( {H}^{m} \) and will no longer consider its supergraph \( {H}^{r} \) . We write \( {H}^{\prime } \mathrel{\text{:=}} f\left( {H}^{m}\right) \) for the corresponding \( T{H}^{m} \) in \( G \) and \( {C}_{i} \mathrel{\text{:=}} f\left( {S}_{i}\right) \) for its concentric cycles, the images of the cycles \( {S}_{i} \) of this \( {H}^{m}\left( {i = 1,\ldots, m}\right) \) .  Fig. 12.5.2. Disjoint copies of \( {H}^{m}\left( {m = 3}\right) \) linked up by a tree in the rest of \( {H}^{r} \) \( e \) \( \sigma : G - e \hookrightarrow S \) Pick an edge \( e = {uv} \) of \( {C}_{1} \), and choose an embedding \( \sigma \) of \( G - e \) in \( S \) . As before, Lemma B. 6 implies that one of the \( \varepsilon + 1 \) disjoint rings \( k \) \( {R}_{{3i} + 2} \) in \( {H}^{m}\left( {i = 0,\ldots ,\varepsilon }\right) ,{R}_{k} \) say, has the property that its hexagons correspond (by \( \sigma \circ f \) ) to circles in \( S \) that bound disjoint open discs there (Figure 12.5.3). Let \( R \supseteq \left( {\sigma \circ f}\right) \left( {R}_{k}\right) \) be the closure in \( S \) of the union of those discs, which is a cylinder in \( S \) . One of its two boundary circles is the image under \( \sigma \) of the cycle \( C \mathrel{\text{:=}} {C}_{k + 1} \) in \( {H}^{\prime } \) to which \( f \) maps the perimeter cycle \( {S}_{k + 1} \) of our special ring \( {R}_{k} \) in \( {H}^{m} \) . \( {H}^{\prime \prime } \) Let \( {H}^{\prime \prime } \mathrel{\text{:=}} f\left( {H}^{k + 1}\right) \subseteq G \), where \( {H}^{k + 1} \) is canonical in \( {H}^{m} \) . Recall that \( {\sigma }^{\prime } \circ f \) maps the hexagons of \( {H}^{k + 1} \) to circles in \( {S}^{\prime } \) bounding disjoint open discs there. The closure in \( {S}^{\prime } \) of the union of these discs is a disc  Fig. 12.5.3. A tree linking up hexagons selected from the rings \( {R}_{2},{R}_{5},{R}_{8}\ldots \) \( {D}^{\prime } \) in \( {S}^{\prime } \), bounded by \( {\sigma }^{\prime }\left( C\right) \) . Deleting a small open disc inside \( {D}^{\prime } \) that does not meet \( {\sigma }^{\prime }\left( G\right) \), we obtain a cylinder \( {R}^{\prime } \subseteq {S}^{\prime } \) that contains \( {\sigma }^{\prime }\left( {H}^{\p

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+ 1} \) of our special ring \( {R}_{k} \) in \( {H}^{m} \) . \( {H}^{\prime \prime } \) Let \( {H}^{\prime \prime } \mathrel{\text{:=}} f\left( {H}^{k + 1}\right) \subseteq G \), where \( {H}^{k + 1} \) is canonical in \( {H}^{m} \) . Recall that \( {\sigma }^{\prime } \circ f \) maps the hexagons of \( {H}^{k + 1} \) to circles in \( {S}^{\prime } \) bounding disjoint open discs there. The closure in \( {S}^{\prime } \) of the union of these discs is a disc  Fig. 12.5.3. A tree linking up hexagons selected from the rings \( {R}_{2},{R}_{5},{R}_{8}\ldots \) \( {D}^{\prime } \) in \( {S}^{\prime } \), bounded by \( {\sigma }^{\prime }\left( C\right) \) . Deleting a small open disc inside \( {D}^{\prime } \) that does not meet \( {\sigma }^{\prime }\left( G\right) \), we obtain a cylinder \( {R}^{\prime } \subseteq {S}^{\prime } \) that contains \( {\sigma }^{\prime }\left( {H}^{\prime \prime }\right) \) . We shall now combine the embeddings \( \sigma : G - e \hookrightarrow S \) and \( {\sigma }^{\prime } : G \hookrightarrow {S}^{\prime } \) to an embedding \( {\sigma }^{\prime \prime } : G \hookrightarrow S \), which will contradict the choice of \( G \) . Let \( \varphi : {\sigma }^{\prime }\left( C\right) \rightarrow \sigma \left( C\right) \) be a homeomorphism between the images of \( C \) in \( {S}^{\prime } \) and in \( S \) that commutes with these embeddings, i.e., is such that \( {\left. \sigma \right| }_{C} = {\left. \left( \varphi \circ {\sigma }^{\prime }\right) \right| }_{C} \) . Then extend this to a homeomorphism \( \varphi : {R}^{\prime } \rightarrow R \) . The idea now is to define \( {\sigma }^{\prime \prime } \) as \( \varphi \circ {\sigma }^{\prime } \) on the part of \( G \) which \( {\sigma }^{\prime } \) maps to \( {D}^{\prime } \) (which includes the edge \( e \) on which \( \sigma \) is undefined), and as \( \sigma \) on the rest of \( G \) (Fig. 12.5.4).  Fig. 12.5.4. Combining \( {\sigma }^{\prime } : G \hookrightarrow {S}^{\prime } \) and \( \sigma : G - e \hookrightarrow S \) to \( {\sigma }^{\prime \prime } : G \hookrightarrow \mathrm{S} \) To make these two partial maps compatible, we start by defining \( {\sigma }^{\prime \prime } \) on \( {\left. C\text{as}\sigma \right| }_{C} = {\left. \left( \varphi \circ {\sigma }^{\prime }\right) \right| }_{C} \) . Next, we define \( {\sigma }^{\prime \prime } \) separately on the components of \( G - C \) . Since \( {\sigma }^{\prime }\left( C\right) \) bounds the disc \( {D}^{\prime } \) in \( {S}^{\prime } \), we know that \( {\sigma }^{\prime } \) maps each component \( J \) of \( G - C \) either entirely to \( {D}^{\prime } \) or entirely to \( {S}^{\prime } \smallsetminus {D}^{\prime } \) . On all the components \( J \) such that \( {\sigma }^{\prime }\left( J\right) \subseteq {D}^{\prime } \), and on all the edges they send to \( G \), we define \( {\sigma }^{\prime \prime } \) as \( \varphi \circ {\sigma }^{\prime } \) . Thus, \( {\sigma }^{\prime \prime } \) embeds these components in \( R \) . Since \( e \in f\left( {H}^{k}\right) = {H}^{\prime \prime } - C \), this includes the component of \( G - C \) that contains \( e \) . It remains to define \( {\sigma }^{\prime \prime } \) on the components of \( G - C \) which \( {\sigma }^{\prime } \) maps to \( {S}^{\prime } \smallsetminus {D}^{\prime } \) . As \( {\sigma }^{\prime }\left( {C}_{k}\right) \subseteq {D}^{\prime } \), these do not meet \( {C}_{k} \) . Since \( \sigma \left( {C \cup {C}_{k}}\right) \) is the frontier of \( R \) in \( S \), this means that \( \sigma \left( J\right) \subseteq S \smallsetminus R \) or \( \sigma \left( J\right) \subseteq R \) for every such component \( J \) . For the component \( {J}_{0} \) of \( G - C \) that contains \( {C}_{k + 2} \) we cannot have \( \sigma \left( {J}_{0}\right) \subseteq R \) : as \( {S}_{k + 2} \cap {R}_{k} = \varnothing \), this would mean that \( \sigma \left( {C}_{k + 2}\right) \) lies in a disc \( D \subseteq R \) corresponding to a face of \( {R}_{k} \), which is impossible since \( {S}_{k + 2} \) sends edges to vertices of \( {S}_{k + 1} \) outside the boundary of that face. We thus have \( \sigma \left( {J}_{0}\right) \subseteq S \smallsetminus R \), and define \( {\sigma }^{\prime \prime } \) as \( \sigma \) on \( {J}_{0} \) and on all the \( {J}_{0} - C \) edges of \( G \) . Next, consider any remaining component \( J \) of \( G - C \) that sends no edge to \( C \) . If \( \sigma \left( J\right) \subseteq S \smallsetminus R \), we define \( {\sigma }^{\prime \prime } \) on \( J \) as \( \sigma \) . If \( \sigma \left( J\right) \subseteq R \), then \( J \) is planar. Since \( J \) sends no edge to \( C \), we can have \( {\sigma }^{\prime \prime } \) map \( J \) to any open disc in \( R \) that has not so far been used by \( {\sigma }^{\prime \prime } \) . It remains to define \( {\sigma }^{\prime \prime } \) on the components \( J \neq {J}_{0} \) of \( G - C \) which \( {\sigma }^{\prime } \) maps to \( {S}^{\prime } \smallsetminus {D}^{\prime } \) and for which \( G \) contains a \( J - C \) edge. Let \( \mathcal{J} \) be the set of all those components \( J \) . We shall group them by the way they attach to \( C \), and define \( {\sigma }^{\prime \prime } \) for these groups in turn. Since \( m \geq k + 2 \), the disc \( {D}^{\prime } \) lies inside a larger disc in \( {S}^{\prime } \), which is the union of \( {D}^{\prime } \) and closed discs \( {D}^{\prime \prime } \) bounded by the images under \( {\sigma }^{\prime } \circ f \) of the hexagons in \( {R}_{k + 1} \) . By definition of \( \mathcal{J} \), the embedding \( {\sigma }^{\prime } \) maps every \( J \in \mathcal{J} \) to such a disc \( {D}^{\prime \prime } \) (Fig. 12.5.5). On the path \( P \) in \( C \) such that \( {\sigma }^{\prime }\left( P\right) = {\sigma }^{\prime }\left( C\right) \cap {D}^{\prime \prime } \) (which is the image under \( f \) of one or two consecutive edges on \( {S}_{k + 1} \) ), let \( {v}_{1},\ldots ,{v}_{n} \) be the vertices with a neighbour in \( {J}_{0} \), in their natural order along \( P \), and write \( {P}_{i} \) for the segment of \( P \) from \( {v}_{i} \) to \( {v}_{i + 1} \) . For any \( {v}_{i} \) with \( 1 < i < n \), pick a \( {v}_{i} - {J}_{0} \) edge and extend it through \( {J}_{0} \) to a path \( Q \) from \( {v}_{i} \) to \( {C}_{k + 2} \) (which exists by definition of \( {J}_{0} \) ); let \( w \) be its first vertex that \( {\sigma }^{\prime } \) maps to the boundary circle of \( {D}^{\prime \prime } \) . By Lemma 4.1.2 applied to \( {\sigma }^{\prime }\left( {{v}_{i}{Qw}}\right) \) and the two arcs joining \( {\sigma }^{\prime }\left( {v}_{i}\right) \) to \( {\sigma }^{\prime }\left( w\right) \) along the boundary circle of \( {D}^{\prime \prime } \), there is no arc through \( {D}^{\prime \prime } \) that links \( {\sigma }^{\prime }\left( {P}_{i - 1}\right) \) to \( {\sigma }^{\prime }\left( {P}_{i}\right) \) but avoids \( {\sigma }^{\prime }\left( {{v}_{i}{Qw}}\right) \) . Hence, every \( J \in \mathcal{J} \) with \( {\sigma }^{\prime }\left( J\right) \subseteq {D}^{\prime \prime } \) has all its neighbours on \( C \) in the same \( {P}_{i} \) , and \( {\sigma }^{\prime } \) maps \( J \) to the face \( {f}_{i} \) of the plane graph \( {\sigma }^{\prime }\left( {G\left\lbrack {{J}_{0} \cup C}\right\rbrack }\right) \cap {D}^{\prime \prime } \) whose boundary contains \( {P}_{i} \) . We shall define \( {\sigma }^{\prime \prime } \) jointly on all those \( J \in \mathcal{J} \) which \( {\sigma }^{\prime } \) maps to this \( {f}_{i} \), for \( i = 1,\ldots, n - 1 \) in turn. To do so, we choose an open disc \( {D}_{i} \) in \( S \smallsetminus R \) that has a boundary circle containing \( \sigma \left( {P}_{i}\right) \) and avoids the image of \( {\sigma }^{\prime \prime } \) as defined until now. Such \( {D}_{i} \) exists in a strip neighbourhood of \( \sigma \left( C\right) \) in \( S \), because components \( {J}^{\prime } \in \mathcal{J} \) attaching to a segment \( {P}_{j} \neq {P}_{i} \) of \( C \) send no edge to \( {P}_{i} \) . Choose a homeomorphism \( {\varphi }_{i} \) from the boundary circle of \( {f}_{i} \) to that of \( {D}_{i} \) so that \( {\left. \sigma \right| }_{{P}_{i}} = {\left. \left( {\varphi }_{i} \circ {\sigma }^{\prime }\right) \right| }_{{P}_{i}} \), and extend this to a homeomorphism \( {\varphi }_{i} \) from the closure of \( {f}_{i} \) in \( {S}^{\prime } \) to the closure of \( {D}_{i} \) in \( S \) . For every \( J \in \mathcal{J} \) with \( {\sigma }^{\prime }\left( J\right) \subseteq {f}_{i} \), and for all \( J - C \) edges of \( G \), define \( {\sigma }^{\prime \prime } \) as \( {\varphi }_{i} \circ {\sigma }^{\prime } \) .  Fig. 12.5.5. Define \( {\sigma }^{\prime \prime } \) jointly for the components \( J,{J}^{\prime } \in \mathcal{J} \) that attach to the same \( {P}_{i} \subseteq C \) \( \left( {1.7.2}\right) \) Proof of Corollary 12.5.3. By their minimality, the graphs in (12.3.7) \( {\mathcal{K}}_{\mathcal{P}\left( S\right) } \) are incomparable under the minor-relation. If their tree-width is (12.4.4) bounded, then \( {\mathcal{K}}_{\mathcal{P}\left( S\right) } \) is well-quasi-ordered by the minor relation (Theorem 12.3.7), and hence must be finite. So assume their tree-width is unbounded, and let \( r \) be as in Lemma 12.5.4. By Theorem 12.4.4, some \( H \in {\mathcal{K}}_{\mathcal{P}\left( S\right) } \) has a grid minor large enough to contain \( {H}^{r} \) . By Proposition 1.7.2, \( {H}^{r} \) is a topological minor of \( H \), contrary to the choice of \( r \) . We finally come to the proof of the graph minor theorem itself. T

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

_0.jpg](images/ecdfe8f9-7805-48fc-b1e8-df5c40de94c0_358_0.jpg) Fig. 12.5.5. Define \( {\sigma }^{\prime \prime } \) jointly for the components \( J,{J}^{\prime } \in \mathcal{J} \) that attach to the same \( {P}_{i} \subseteq C \) \( \left( {1.7.2}\right) \) Proof of Corollary 12.5.3. By their minimality, the graphs in (12.3.7) \( {\mathcal{K}}_{\mathcal{P}\left( S\right) } \) are incomparable under the minor-relation. If their tree-width is (12.4.4) bounded, then \( {\mathcal{K}}_{\mathcal{P}\left( S\right) } \) is well-quasi-ordered by the minor relation (Theorem 12.3.7), and hence must be finite. So assume their tree-width is unbounded, and let \( r \) be as in Lemma 12.5.4. By Theorem 12.4.4, some \( H \in {\mathcal{K}}_{\mathcal{P}\left( S\right) } \) has a grid minor large enough to contain \( {H}^{r} \) . By Proposition 1.7.2, \( {H}^{r} \) is a topological minor of \( H \), contrary to the choice of \( r \) . We finally come to the proof of the graph minor theorem itself. The complete proof would still fill a book or two, but we are well equipped now to get a good understanding of its main ideas and overall structure. For background on surfaces, we once more refer to Appendix B. Proof of the graph minor theorem (sketch). We have to show that every infinite sequence \[ {G}_{0},{G}_{1},{G}_{2},\ldots \] of finite graphs contains a good pair: two graphs \( {G}_{i} \preccurlyeq {G}_{j} \) with \( i < j \) . We may assume that \( {G}_{0} \npreceq {G}_{i} \) for all \( i \geq 1 \), since \( {G}_{0} \) forms a good pair with any graph \( {G}_{i} \) of which it is a minor. Thus all the graphs \( {G}_{1},{G}_{2},\ldots \) lie in \( {\operatorname{Forb}}_{ \preccurlyeq }\left( {G}_{0}\right) \), and we may use the structure common to these graphs in our search for a good pair. We have already seen how this works when \( {G}_{0} \) is planar: then the graphs in \( {\operatorname{Forb}}_{ \preccurlyeq }\left( {G}_{0}\right) \) have bounded tree-width (Theorem 12.4.3) and are therefore well-quasi-ordered by Theorem 12.3.7. In general, we need only consider the cases of \( {G}_{0} = {K}^{n} \) : since \( {G}_{0} \preccurlyeq {K}^{n} \) for \( n \mathrel{\text{:=}} \left| {G}_{0}\right| \), we may assume that \( {K}^{n} \npreceq {G}_{i} \) for all \( i \geq 1 \) . The proof now follows the same lines as above: again the graphs in \( {\operatorname{Forb}}_{ \preccurlyeq }\left( {K}^{n}\right) \) can be characterized by their tree-decompositions, and again their tree structure helps, as in Kruskal's theorem, with the proof that they are well-quasi-ordered. The parts in these tree-decompositions are no longer restricted in terms of order now, but they are constrained in more subtle structural terms. Roughly speaking, for every \( n \) there exists a finite set \( \mathcal{S} \) of surfaces such that every graph without a \( {K}^{n} \) minor has a tree-decomposition into parts each 'nearly' embeddable in one of the surfaces \( S \in \mathcal{S} \) ; see Theorem 12.4.11. By a generalization of Theorem 12.3.7 and hence of Kruskal's theorem-it now suffices, essentially, to prove that the set of all the parts in these tree-decompositions is well-quasi-ordered: then the graphs decomposing into these parts are well-quasi-ordered, too. Since \( \mathcal{S} \) is finite, every infinite sequence of such parts has an infinite subsequence whose members are all (nearly) embeddable in the same surface \( S \in \mathcal{S} \) . Thus all we have to show is that, given any surface \( S \), all the graphs embeddable in \( S \) are well-quasi-ordered by the minor relation. This is shown by induction on the Euler genus of \( S \), using the same approach as before: if \( {H}_{0},{H}_{1},{H}_{2},\ldots \) is an infinite sequence of graphs embeddable in \( S \), we may assume that none of the graphs \( {H}_{1},{H}_{2},\ldots \) contains \( {H}_{0} \) as a minor. If \( S = {S}^{2} \) we are back in the case that \( {H}_{0} \) is planar, so the induction starts. For the induction step we now assume that \( S \neq {S}^{2} \) . Again, the exclusion of \( {H}_{0} \) as a minor constrains the structure of the graphs \( {H}_{1},{H}_{2},\ldots \), this time topologically: each \( {H}_{i} \) with \( i \geq 1 \) has an embedding in \( S \) which meets some circle \( {C}_{i} \subseteq S \) that does not bound a disc in \( S \) in no more than a bounded number of vertices (and no edges), say in \( {X}_{i} \subseteq V\left( {H}_{i}\right) \) . (The bound on \( \left| {X}_{i}\right| \) depends on \( {H}_{0} \) , but not on \( {H}_{i} \) .) Cutting along \( {C}_{i} \) and capping the hole(s), we obtain one or two new surfaces of smaller Euler genus. If the cut produces only one new surface \( {S}_{i} \), then our embedding of \( {H}_{i} - {X}_{i} \) still counts as a near-embedding of \( {H}_{i} \) in \( {S}_{i} \) (since \( {X}_{i} \) is small). If this happens for infinitely many \( i \), then infinitely many of the surfaces \( {S}_{i} \) are also the same, and the induction hypothesis gives us a good pair among the corresponding graphs \( {H}_{i} \) . On the other hand, if we get two surfaces \( {S}_{i}^{\prime } \) and \( {S}_{i}^{\prime \prime } \) for infinitely many \( i \) (without loss of generality the same two surfaces), then \( {H}_{i} \) decomposes accordingly into subgraphs \( {H}_{i}^{\prime } \) and \( {H}_{i}^{\prime \prime } \) embedded in these surfaces, with \( V\left( {{H}_{i}^{\prime } \cap {H}_{i}^{\prime \prime }}\right) = {X}_{i} \) . The set of all these subgraphs taken together is again well-quasi-ordered by the induction hypothesis, and hence so are the pairs \( \left( {{H}_{i}^{\prime },{H}_{i}^{\prime \prime }}\right) \) by Lemma 12.1.3. Using a sharpening of the lemma that takes into account not only the graphs \( {H}_{i}^{\prime } \) and \( {H}_{i}^{\prime \prime } \) themselves but also how \( {X}_{i} \) lies inside them, we finally obtain indices \( i, j \) not only with \( {H}_{i}^{\prime } \preccurlyeq {H}_{j}^{\prime } \) and \( {H}_{i}^{\prime \prime } \preccurlyeq {H}_{j}^{\prime \prime } \), but also such that these minor embeddings extend to the desired minor embedding of \( {H}_{i} \) in \( {H}_{j} \) -completing the proof of the graph minor theorem. The graph minor theorem does not extend to graphs of arbitrary cardinality, but it might extend to countable graphs. Whether or not it does appears to be a difficult problem. It may be related to the following intriguing conjecture, which easily implies the graph minor theorem for finite graphs (Exercise 44). Call a graph \( H \) a proper minor of \( G \) if \( G \) contains an \( {MH} \) with at least one non-trivial branch set. ## Self-minor conjecture. (Seymour 1980s) Every countably infinite graph is a proper minor of itself. In addition to its impact on 'pure' graph theory, the graph minor theorem has had far-reaching algorithmic consequences. Using their structure theorem for the graphs in \( {\operatorname{Forb}}_{ \preccurlyeq }\left( {K}^{n}\right) \), Theorem 12.4.11, Robertson and Seymour have shown that testing for any fixed minor is ’fast’: for every graph \( H \) there is a polynomial-time algorithm \( {}^{7} \) that decides whether or not the input graph contains \( H \) as a minor. By the minor theorem, then, every minor-closed graph property \( \mathcal{P} \) can be decided in polynomial (even cubic) time: if \( {\mathcal{K}}_{\mathcal{P}} = \left\{ {{H}_{1},\ldots ,{H}_{k}}\right\} \) is the corresponding set of forbidden minors, then testing a graph \( G \) for membership in \( \mathcal{P} \) reduces to testing the \( k \) assertions \( {H}_{i} \preccurlyeq G \) . The following example gives an indication of how deeply this algorithmic corollary affects the complexity theory of graph algorithms. Let us call a graph knotless if it can be embedded in \( {\mathbb{R}}^{3} \) so that none of its cycles forms a non-trivial knot. Before the graph minor theorem, it was an open problem whether knotlessness is decidable, that is, whether any algorithm exists (no matter how slow) that decides for any given graph whether or not that graph is knotless. To this day, no such algorithm is known. The property of knotlessness, however, is easily 'seen' to be closed under taking minors: contracting an edge of a graph embedded in 3-space will not create a knot where none had been before. Hence, by the minor theorem, there exists an algorithm that decides knotlessness even in polynomial (cubic) time! However spectacular such unexpected solutions to long-standing problems may be, viewing the graph minor theorem merely in terms of its corollaries will not do it justice. At least as important are the techniques developed for its proof, the various ways in which minors are handled or constructed. Most of these have not even been touched upon here, yet they seem set to influence the development of graph theory for many years to come. --- 7 indeed a cubic one although with a typically enormous constant depending on \( H \) --- ## Exercises 1. \( {}^{ - } \) Let \( \leq \) be a quasi-ordering on a set \( X \) . Call two elements \( x, y \in X \) equivalent if both \( x \leq y \) and \( y \leq x \) . Show that this is indeed an equivalence relation on \( X \), and that \( \leq \) induces a partial ordering on the set of equivalence classes. 2. Let \( \left( {A, \leq }\right) \) be a quasi-ordering. For subsets \( X \subseteq A \) write \[ {\operatorname{Forb}}_{ \leq }\left( X\right) \mathrel{\text{:=}} \{ a \in A \mid a \ngeq x\text{ for all }x \in X\} . \] Show that \( \leq \) is a well-quasi-ordering on \( A \) if and only if every subset \( B \subseteq A \) that is closed under \( \geq \) (i.e. such that \( x \leq y \in B \Rightarrow x \in B \) ) can be written as \( B = {\operatorname{Forb}}_{ \leq }\left( X\right) \) with finite \( X \) . 3. Prove Proposition 12.1.1 and Corollary 12.1.2 directly, without using Ramsey's theorem. 4. Given a quasi

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pically enormous constant depending on \( H \) --- ## Exercises 1. \( {}^{ - } \) Let \( \leq \) be a quasi-ordering on a set \( X \) . Call two elements \( x, y \in X \) equivalent if both \( x \leq y \) and \( y \leq x \) . Show that this is indeed an equivalence relation on \( X \), and that \( \leq \) induces a partial ordering on the set of equivalence classes. 2. Let \( \left( {A, \leq }\right) \) be a quasi-ordering. For subsets \( X \subseteq A \) write \[ {\operatorname{Forb}}_{ \leq }\left( X\right) \mathrel{\text{:=}} \{ a \in A \mid a \ngeq x\text{ for all }x \in X\} . \] Show that \( \leq \) is a well-quasi-ordering on \( A \) if and only if every subset \( B \subseteq A \) that is closed under \( \geq \) (i.e. such that \( x \leq y \in B \Rightarrow x \in B \) ) can be written as \( B = {\operatorname{Forb}}_{ \leq }\left( X\right) \) with finite \( X \) . 3. Prove Proposition 12.1.1 and Corollary 12.1.2 directly, without using Ramsey's theorem. 4. Given a quasi-ordering \( \left( {X, \leq }\right) \) and subsets \( A, B \subseteq X \), write \( A{ \leq }^{\prime }B \) if there exists an order preserving injection \( f : A \rightarrow B \) with \( a \leq f\left( a\right) \) for all \( a \in A \) . Does Lemma 12.1.3 still hold if the quasi-ordering considered for \( {\left\lbrack X\right\rbrack }^{ < \omega } \) is \( { \leq }^{\prime } \) ? 5. \( {}^{ - } \) Show that the relation \( \leq \) between rooted trees defined in the text is indeed a quasi-ordering. 6. Show that the finite trees are not well-quasi-ordered by the subgraph relation. 7. The last step of the proof of Kruskal's theorem considers a 'topological' embedding of \( {T}_{m} \) in \( {T}_{n} \) that maps the root of \( {T}_{m} \) to the root of \( {T}_{n} \) . Suppose we assume inductively that the trees of \( {A}_{m} \) are embedded in the trees of \( {A}_{n} \) in the same way, with roots mapped to roots. We thus seem to obtain a proof that the finite rooted trees are well-quasi-ordered by the subgraph relation, even with roots mapped to roots. Where is the error? 8. Are the connected finite graphs well-quasi-ordered by contraction alone (i.e. by taking minors without deleting edges or vertices)? 9. \( {}^{ + } \) Relax the minor relation by not insisting that branch sets be connected. Show that the finite graphs are well-quasi-ordered by this relation. 10. \( {}^{ + } \) Show that the finite graphs are not well-quasi-ordered by the topological minor relation. 11. \( {}^{ + } \) Given \( k \in \mathbb{N} \), is the class \( \left\{ {G \mid G \nsupseteq {P}^{k}}\right\} \) well-quasi-ordered by the subgraph relation? 12. \( {}^{ - } \) Let \( G \) be a graph, \( T \) a tree, and \( \mathcal{V} = {\left( {V}_{t}\right) }_{t \in T} \) a family of subsets of \( V\left( G\right) \) . Show that \( \left( {T,\mathcal{V}}\right) \) is a tree-decomposition of \( G \) if and only if (i) for every \( v \in V\left( G\right) \) the set \( {T}_{v} \mathrel{\text{:=}} \left\{ {t \mid v \in {V}_{t}}\right\} \) induces a subtree of \( T \) ; (ii) \( {T}_{u} \cap {T}_{v} \neq \varnothing \) for every edge \( {uv} \) of \( G \) . 13. \( {}^{ - } \) Show that every graph \( G \) has a tree-decomposition of width \( \operatorname{tw}\left( G\right) \) in which no part contains another. 14. Show that a graph has tree-width at most 1 if and only if it is a forest. 15. Let \( G \) be a graph, \( T \) a set, and \( {\left( {V}_{t}\right) }_{t \in T} \) a family of subsets of \( V\left( G\right) \) satisfying (T1) and (T2) from the definition of a tree-decomposition. Show that there exists a tree on \( T \) that makes (T3) true if and only if there exists an enumeration \( {t}_{1},\ldots ,{t}_{n} \) of \( T \) such that for every \( k = 2,\ldots, n \) there is a \( j < k \) satisfying \( {V}_{{t}_{k}} \cap \mathop{\bigcup }\limits_{{i < k}}{V}_{{t}_{i}} \subseteq {V}_{{t}_{j}} \) . (The new condition tends to be more convenient to check than (T3). It can help, for example, with the construction of a tree-decomposition into a given set of parts.) 16. Prove the following converse of Lemma 12.3.1: if \( \left( {T,\mathcal{V}}\right) \) satisfies condition (T1) and the statement of the lemma, then \( \left( {T,\mathcal{V}}\right) \) is a tree-decomposition of \( G \) . 17. Can the tree-width of a subdivision of a graph \( G \) be smaller than \( \operatorname{tw}\left( G\right) \) ? Can it be larger? 18. \( {}^{ + } \) Show that if a graph has circumference \( k \neq 0 \), then its tree-width is at most \( k - 1 \) . 19. Call two separations \( \left\{ {{U}_{1},{U}_{2}}\right\} \) and \( \left\{ {{W}_{1},{W}_{2}}\right\} \) of \( G \) compatible if we can choose \( i, j \in \{ 1,2\} \) so that \( {U}_{i} \subseteq {W}_{j} \) and \( {U}_{3 - i} \supseteq {W}_{3 - j} \) . (i) Show that the separations \( {S}_{e} \mathrel{\text{:=}} \left\{ {{U}_{1},{U}_{2}}\right\} \) in Lemma 12.3.1 are compatible for different choices of the edge \( e = {t}_{1}{t}_{2} \in T \) . \( {\text{(ii)}}^{ + } \) Conversely, show that given a set \( \mathcal{S} \) of compatible separations of \( G \) there is a tree-decomposition \( \left( {\mathcal{V}, T}\right) \) of \( G \) such that \( \mathcal{S} = \) \( \left\{ {{S}_{e} \mid e \in E\left( T\right) }\right\} \) 20. \( {}^{ + } \) Show that every 2-connected graph has a tree-decomposition \( \left( {T,{\left( {V}_{t}\right) }_{t \in T}}\right) \) such that \( \left| {{V}_{t} \cap {V}_{{t}^{\prime }}}\right| = 2 \) for every edge \( t{t}^{\prime } \in T \) and all torsos are either 3-connected or a cycle. Conversely, show that every graph with such a tree-decomposition is 2-connected. (Hint. Try a tree-decomposition defined, as in Exercise 19 (ii), by the set of all 2-separations (separations \( \left\{ {{U}_{1},{U}_{2}}\right. \) \} such that \( \left. {\left| {{U}_{1} \cap {U}_{2}}\right| = 2}\right) \) that are compatible with all other 2-separations.) 21. Apply Theorem 12.3.9 to show that the \( k \times k \) grid has tree-width at least \( k \), and find a tree-decomposition of width exactly \( k \) . 22. Let \( \mathcal{B} \) be a maximum-order bramble in a graph \( G \) . Show that every minimum-width tree-decomposition of \( G \) has a unique part covering \( \mathcal{B} \) . 23. \( {}^{ + } \) In the second half of the proof of Theorem 12.3.9, let \( {H}^{\prime } \) be the union of \( H \) and the paths \( {P}_{1},\ldots ,{P}_{\ell } \), let \( {H}^{\prime \prime } \) be the graph obtained from \( {H}^{\prime } \) by contracting each \( {P}_{i} \), and let \( \left( {T,{\left( {W}_{t}^{\prime \prime }\right) }_{t \in T}}\right) \) be the tree-decomposition induced on \( {H}^{\prime \prime } \) (as in Lemma 12.3.3) by the decomposition that \( \left( {T,{\left( {V}_{t}\right) }_{t \in T}}\right) \) induces on \( {H}^{\prime } \) . Is this, after the obvious identification of \( {H}^{\prime \prime } \) with \( H \), the same decomposition as the one used in the proof, i.e. is \( {W}_{t}^{\prime \prime } = {W}_{t} \) for all \( t \in T \) ? 24. \( {}^{ - } \) Show that any graph with a simplicial tree-decomposition into \( k \) - colourable parts is itself \( k \) -colourable. 25. Let \( \mathcal{H} \) be a set of graphs, and let \( G \) be constructed recursively from elements of \( \mathcal{H} \) by pasting along complete subgraphs. Show that \( G \) has a simplicial tree-decomposition into elements of \( \mathcal{H} \) . 26. Use the previous exercise to show that \( G \) has no \( {K}^{5} \) minor if and only if \( G \) has a tree-decomposition in which every torso is either planar or a copy of the Wagner graph \( W \) (Figure 7.3.1). 27. \( {}^{ + } \) Call a graph irreducible if it is not separated by any complete subgraph. Every finite graph \( G \) can be decomposed into irreducible induced subgraphs, as follows. If \( G \) has a separating complete subgraph \( S \), then decompose \( G \) into proper induced subgraphs \( {G}^{\prime } \) and \( {G}^{\prime \prime } \) with \( G = {G}^{\prime } \cup {G}^{\prime \prime } \) and \( {G}^{\prime } \cap {G}^{\prime \prime } = S \) . Then decompose \( {G}^{\prime } \) and \( {G}^{\prime \prime } \) in the same way, and so on, until all the graphs obtained are irreducible. By Exercise 25, \( G \) has a simplicial tree-decomposition into these irreducible subgraphs. Show that they are uniquely determined if the complete separators were all chosen minimal. 28. If \( \mathcal{F} \) is a family of sets, then the graph \( G \) on \( \mathcal{F} \) with \( {XY} \in E\left( G\right) \Leftrightarrow \) \( X \cap Y \neq \varnothing \) is called the intersection graph of \( \mathcal{F} \) . Show that a graph is chordal if and only if it is isomorphic to the intersection graph of a family of (vertex sets of) subtrees of a tree. 29. Show that a graph has a path-decomposition into complete graphs if and only if it is isomorphic to an interval graph. (Interval graphs are defined in Ex. 39, Ch. 5.) 30. (continued) The path-width \( \operatorname{pw}\left( G\right) \) of a graph \( G \) is the least width of a path-decomposition of \( G \) . Prove the following analogue of Corollary 12.3.12 for path-width: every graph \( G \) satisfies \( \operatorname{pw}\left( G\right) = \min \omega \left( H\right) - 1 \), where the minimum is taken over all interval graphs \( H \) containing \( G \) . 31. \( {}^{ + } \) Do trees have unbounded path-width? 32. \( {}^{ - } \) Let \( \mathcal{P} \) be a minor-closed graph property. Show that strengthening the notion of a minor (for example, to that of topological minor) increases the set of forbidden minors required to characterize \( \mathcal{P} \) . 33. Deduce from the minor theorem that every minor-closed property can be expressed by forbidding finitely many topological minors. Is the same true for every property that is closed under taking topological minors? 34. \( {}^{ - } \) Show that every horizontal p

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are defined in Ex. 39, Ch. 5.) 30. (continued) The path-width \( \operatorname{pw}\left( G\right) \) of a graph \( G \) is the least width of a path-decomposition of \( G \) . Prove the following analogue of Corollary 12.3.12 for path-width: every graph \( G \) satisfies \( \operatorname{pw}\left( G\right) = \min \omega \left( H\right) - 1 \), where the minimum is taken over all interval graphs \( H \) containing \( G \) . 31. \( {}^{ + } \) Do trees have unbounded path-width? 32. \( {}^{ - } \) Let \( \mathcal{P} \) be a minor-closed graph property. Show that strengthening the notion of a minor (for example, to that of topological minor) increases the set of forbidden minors required to characterize \( \mathcal{P} \) . 33. Deduce from the minor theorem that every minor-closed property can be expressed by forbidding finitely many topological minors. Is the same true for every property that is closed under taking topological minors? 34. \( {}^{ - } \) Show that every horizontal path in the \( k \times k \) grid is externally \( k \) - connected in that grid. 35. \( {}^{ + } \) Show that the tree-width of a graph is large if and only if it contains a large externally \( k \) -connected set of vertices, with \( k \) large. For example, show that graphs of tree-width \( < k \) contain no externally \( \left( {k + 1}\right) \) - connected set of \( {3k} \) vertices, and that graphs containing no externally \( \left( {k + 1}\right) \) -connected set of \( {3k} \) vertices have tree-width \( < {4k} \) . 36. \( {}^{ + } \) (continued) Find an \( \mathbb{N} \rightarrow {\mathbb{N}}^{2} \) function \( k \mapsto \left( {h,\ell }\right) \) such that every graph with an externally \( \ell \) -connected set of \( h \) vertices contains a bramble of order at least \( k \) . Deduce the weakening of Theorem 12.3.9 that, given \( k \), every graph of large enough tree-width contains a bramble of order at least \( k \) . A tangle of order \( k \in \mathbb{N} \) in a graph \( G = \left( {V, E}\right) \) is a set \( \mathcal{T} \) of ordered pairs \( \left( {A, B}\right) \) of subsets of \( V \) satisfying the following conditions. ( \( \mathcal{I}1 \) ) For every \( \left( {A, B}\right) \in \mathcal{T} \), the 2-set \( \{ A, B\} \) is a separation in \( G \) or order \( < k \) . (T2) For every separation \( \{ A, B\} \) of order \( < k \) in \( G \), at least one of \( \left( {A, B}\right) \) , \( \left( {B, A}\right) \) is an element of \( \mathcal{T} \) . (T3) If \( \left( {{A}_{1},{B}_{1}}\right) ,\left( {{A}_{2},{B}_{2}}\right) ,\left( {{A}_{3},{B}_{3}}\right) \in \mathcal{T} \) then \( {A}_{1} \cup {A}_{2} \cup {A}_{3} \neq V \) . ( \( \mathcal{T}4 \) ) No \( \left( {A, B}\right) \in \mathcal{T} \) is such that \( A = V \) . 37. Deduce from Exercise 35 that every graph of tree-width at least \( {4k} \) has a tangle of order \( k \) . 38. Extend Corollary 12.4.10 as follows. Let \( H \) be a connected planar graph, let \( \mathcal{X} \) be any set of connected graphs including \( H \), and let \( \mathcal{H} \mathrel{\text{:=}} \) \( \{ {MX} \mid X \in \mathcal{X}\} \) . Show that \( \mathcal{H} \) has the Erdős-Pósa property, witnessed by the same function \( f \) as defined in the proof of Corollary 12.4.10. Explain how it is possible that \( f \) depends on \( H \) but not on any of the other graphs in \( \mathcal{X} \) . 39. \( {}^{ + } \) Show that, for every non-planar graph \( H \), the class \( {MH} \) fails to have the Erdős-Pósa property. (Hint. Embed \( H \) in a surface \( S \), and consider only graphs embedded in \( S \) .) 40. \( {}^{ + } \) Extend Corollary 12.4.10 to disconnected graphs \( H \), or find a counterexample. 41. \( {}^{ + } \) Show that the four ingredients to the structure of the graphs in \( {\operatorname{Forb}}_{ \preccurlyeq }\left( {K}^{n}\right) \) as described in Theorem 12.4.11-tree-decomposition, an apex set \( X \), genus, and vortices \( {H}_{1},\ldots ,{H}_{k} \) -are all needed to capture all the graphs in \( {\operatorname{Forb}}_{ \preccurlyeq }\left( {K}^{n}\right) \) . More precisely, find examples of graphs in \( {\operatorname{Forb}}_{ \preccurlyeq }\left( {K}^{n}\right) \) showing that Theorem 12.4.11 becomes false if we require in addition that the tree-decomposition has only one part, or that \( X \) is always empty, or that \( S \) is always the sphere, or that \( {H}_{1},\ldots ,{H}_{k} \) are always empty. No exact proofs are required. 42. Without using the minor theorem, show that the chromatic number of the graphs in any \( \preccurlyeq \) -antichain is bounded. 43. Let \( {S}_{g} \) denote the surface obtained from the sphere by adding \( g \) handles. Find a lower bound for \( \left| {\mathcal{K}}_{\mathcal{P}\left( S\right) }\right| \) in terms of \( g \) . (Hint. The smallest \( g \) such that a given graph can be embedded in \( {S}_{g} \) is its orientable genus. Use the theorem that the orientable genus of a graph is equal to the sum of the genera of its blocks.) 44. Deduce the graph minor theorem from the self-minor conjecture. 45. Prove Theorem 12.4.13, assuming that \( G \) has a normal spanning tree. 46. Let \( G \) be a locally finite graph obtained from the \( \mathbb{Z} \times \mathbb{Z} \) grid \( H \) by adding an infinite set of edges \( {xy} \) with \( {d}_{H}\left( {x, y}\right) \) unbounded. Show that \( G \succcurlyeq {K}^{{\aleph }_{0}} \) . Can you do the same if the distances \( {d}_{H}\left( {x, y}\right) \) are bounded (but at least 3)? 47. Is the infinite \( \mathbb{Z} \times \mathbb{Z} \) grid a minor of the \( \mathbb{Z} \times \mathbb{N} \) grid? Is the latter a minor of the \( \mathbb{N} \times \mathbb{N} \) grid? 48. \( {}^{ + } \) Extend Proposition 12.3.11 to infinite graphs not containing an infinite complete subgraph. 49. Using the previous exercise, prove that if every finite subgraph of \( G \) has tree-width less than \( k \in \mathbb{N} \) then so does \( G \) : (i) for countable \( G \), using the infinity lemma; \( {\text{(ii)}}^{ + } \) for arbitrary \( G \), using Zorn’s lemma. 50. Show that no assumption of large finite connectivity can ensure that a countable graph has a \( {K}^{r} \) minor when \( r \geq 5 \) . However, using the previous exercise show that sufficiently large finite connectivity forces any given planar minor. ## Notes Kruskal's theorem on the well-quasi-ordering of finite trees was first published in J.A. Kruskal, Well-quasi ordering, the tree theorem, and Vászonyi's conjecture, Trans. Amer. Math. Soc. 95 (1960), 210-225. Our proof is due to Nash-Williams, who introduced the versatile proof technique of choosing a 'minimal bad sequence'. This technique was also used in our proof of Higman's Lemma 12.1.3. Nash-Williams generalized Kruskal's theorem to infinite graphs. This extension is much more difficult than the finite case. Its proof introduces as a tool the notion of better-quasi-ordering, a concept that has profoundly influenced well-quasi-ordering theory. The graph minor theorem is false for uncountable graphs; this was shown by R. Thomas, A counterexample to 'Wagner's conjecture' for infinite graphs, Math. Proc. Camb. Phil. Soc. 103 (1988), 55- 57. Whether or not the countable graphs are well-quasi-ordered as minors, and whether the finite (or the countable) graphs are better-quasi-ordered as minors, are related questions that remain wide open. Both are related also to the self-minor conjecture. This, too, was originally intended to include graphs of arbitrary cardinality, but was disproved for uncountable graphs by B. Oporowski, A counterexample to Seymour’s self-minor conjecture, J. Graph Theory 14 (1990), 521–524. The notions of tree-decomposition and tree-width were first introduced (under different names) by R. Halin, \( S \) -functions for graphs, J. Geometry 8 (1976), 171-186. Among other things, Halin showed that grids can have arbitrarily large tree-width. Robertson & Seymour reintroduced the two concepts, apparently unaware of Halin's paper, with direct reference to K. Wagner, Über eine Eigenschaft der ebenen Komplexe, Math. Ann. 114 (1937), 570- 590. (This is the seminal paper that introduced simplicial tree-decompositions to prove Theorem 7.3.4; cf. Exercise 26.) Simplicial tree-decompositions are treated in depth in R. Diestel, Graph Decompositions, Oxford University Press 1990. Robertson & Seymour usually refer to the graph minor theorem as Wagner's conjecture. Wagner did indeed discuss this problem in the 1960s with his then students, Halin and Mader, and it is not unthinkable that one of them conjectured a positive solution. Wagner himself always insisted that he did not-even after the graph minor theorem had been proved. Robertson & Seymour's proof of the graph minor theorem is given in the numbers IV-VII, IX-XII and XIV-XX of their series of over 20 papers under the common title of Graph Minors, most of which appeared in the Journal of Combinatorial Theory, Series B, between 1983 and 2004. Of their theorems cited in this chapter, Theorem 12.3.7 is from Graph Minors IV, Theorems 12.4.3 and 12.4.4 are from Graph Minors V, and Theorem 12.4.11 is from Graph Minors XVI. Our short proof of Theorems 12.4.3 and 12.4.4 is from R. Diestel, K.Yu. Gorbunov, T.R. Jensen & C. Thomassen, Highly connected sets and the excluded grid theorem, J. Combin. Theory B 75 (1999), 61-73. Theorem 12.3.9 is due to P.D. Seymour & R. Thomas, Graph searching and a min-max theorem for tree-width, J. Combin. Theory B 58 (1993), 22-33. Our proof is a simplification of the original proof. B.A. Reed gives an instructive introductory survey on tree-width and graph minors, including some algorithmic aspects, in (R.A. Bailey, ed) Surveys in Combinatorics 1997, Cambridge University Press 1997, 87-162. Reed also introduced the term 'bramble'; in Seymour & Thomas's paper, brambles are called 'screens'. The obstructions to small tree-width actually used in the proof of the graph minor

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B, between 1983 and 2004. Of their theorems cited in this chapter, Theorem 12.3.7 is from Graph Minors IV, Theorems 12.4.3 and 12.4.4 are from Graph Minors V, and Theorem 12.4.11 is from Graph Minors XVI. Our short proof of Theorems 12.4.3 and 12.4.4 is from R. Diestel, K.Yu. Gorbunov, T.R. Jensen & C. Thomassen, Highly connected sets and the excluded grid theorem, J. Combin. Theory B 75 (1999), 61-73. Theorem 12.3.9 is due to P.D. Seymour & R. Thomas, Graph searching and a min-max theorem for tree-width, J. Combin. Theory B 58 (1993), 22-33. Our proof is a simplification of the original proof. B.A. Reed gives an instructive introductory survey on tree-width and graph minors, including some algorithmic aspects, in (R.A. Bailey, ed) Surveys in Combinatorics 1997, Cambridge University Press 1997, 87-162. Reed also introduced the term 'bramble'; in Seymour & Thomas's paper, brambles are called 'screens'. The obstructions to small tree-width actually used in the proof of the graph minor theorem are not brambles of large order but tangles; see Exercise 37. Tangles are more powerful than brambles and well worth studying. See Graph Minors X or Reed's survey for an introduction to tangles and their relation to brambles and tree-decompositions. Theorem 12.3.10 is due to R. Thomas, A Menger-like property of tree-width; the finite case, J. Combin. Theory B 48 (1990), 67-76. For a short proof see P. Bellenbaum & R. Diestel, Two short proofs concerning tree-decompositions, Combinatorics, Probability and Computing 11 (2002), 541-547. The Kuratowski set for the graphs of tree-width \( < 4 \) have been determined by S. Arnborg, D.G. Corneil and A. Proskurowski, Forbidden minors characterization of partial 3-trees, Discrete Math. 80 (1990), 1-19. They are: \( {K}^{5} \), the octahedron \( {K}_{2,2,2} \), the 5-prism \( {C}^{5} \times {K}^{2} \), and the Wagner graph \( W \) . As a forerunner to Theorem 12.4.3, Robertson & Seymour proved its following analogue for path-width (Graph Minors I): excluding a graph \( H \) as a minor bounds the path-width of a graph if and only if \( H \) is a forest. A short proof of this result, with optimal bounds, can be found in the first edition of this book, or in R. Diestel, Graph Minors I: a short proof of the path width theorem, Combinatorics, Probability and Computing 4 (1995), 27-30. The Kuratowski set \( {\mathcal{K}}_{\mathcal{P}\left( S\right) } \) for a given surface \( S \) has been determined explicitly for only one surface other than the sphere, the projective plane. It consists of 35 forbidden minors; see D. Archdeacon, A Kuratowski theorem for the projective plane, J. Graph Theory 5 (1981), 243-246. It is not difficult to show that \( \left| {\mathcal{K}}_{\mathcal{P}\left( S\right) }\right| \) grows rapidly with the genus of \( S \) (Exercise 43). An upper bound is given in P.D. Seymour, A bound on the excluded minors for a surface, J. Combin. Theory B (to appear). A survey of finite forbidden minor theorems is given in Chapter 6.1 of R. Diestel, Graph Decompositions, Oxford University Press 1990. More recent developments are surveyed in R. Thomas, Recent excluded minor theorems, in (J.D.Lamb & D.A.Preece, eds) Surveys in Combinatorics 1999, Cambridge University Press 1999, 201-222. A survey of infinite forbidden minor theorems was given by N. Robertson, P.D. Seymour & R. Thomas, Excluding infinite minors, Discrete Math. 95 (1991), 303-319. The existence of normal spanning trees for graphs with no topological \( {K}^{{\aleph }_{0}} \) minor was proved by R.Halin, Simplicial decompositions of infinite graphs, in: (B. Bollobás, ed.) Advances in Graph Theory, Annals of Discrete Mathematics 3, North-Holland 1978. Its strengthening, part (iii) of Theorem 12.4.13, was observed in R.Diestel, The depth-first search tree structure of \( T{K}_{{\aleph }_{0}} \) -free graphs, \( J \) . Combin. Theory B 61 (1994),260-262. Part (iii) easily implies part (ii), which had been proved independently by N. Robertson, P.D. Seymour & R. Thomas, Excluding infinite clique subdivisions, Trans. Amer. Math. Soc. 332 (1992), 211-223. Theorem 12.4.12 was proved in R. Diestel & R. Thomas, Excluding a countable clique, J. Com-bin. Theory B 76 (1999), 41-67; the proof builds on the main result of N. Robertson, P.D. Seymour & R. Thomas, Excluding infinite clique minors, Mem. Amer. Math. Soc. 118 (1995). Our proof of the 'generalized Kuratowski theorem', Corollary 12.5.3, was inspired by J. Geelen, B. Richter & G. Salazar, Embedding grids in surfaces, Europ. J. Combinatorics 25 (2004), 785-792. An alternative proof, which bypasses Theorem 12.3.7 by proving directly that the graphs in \( {\mathcal{K}}_{\mathcal{P}\left( S\right) } \) have bounded order, is given by B. Mohar & C. Thomassen, Graphs on Surfaces, Johns Hopkins University Press 2001. Mohar (see there) also developed a set of algorithms, one for each surface, that decide embeddability in that surface in linear time. As a corollary, he obtains an independent and constructive proof of Corollary 12.5.3. For every graph \( X \), Graph Minors XIII gives an explicit algorithm that decides in cubic time for every input graph \( G \) whether \( X \preccurlyeq G \) . The constants in the cubic polynomials bounding the running time of these algorithms depend on \( X \) but are constructively bounded from above. For an overview of the algorithmic implications of the Graph Minors series, see Johnson's NP-completeness column in J. Algorithms 8 (1987), 285-303. The concept of a 'good characterization' of a graph property was first suggested by J. Edmonds, Minimum partition of a matroid into independent subsets, J. Research of the National Bureau of Standards (B) 69 (1965) 67-72. In the language of complexity theory, a characterization is good if it specifies two assertions about a graph such that, given any graph \( G \), the first assertion holds for \( G \) if and only if the second fails, and such that each assertion, if true for \( G \), provides a certificate for its truth. Thus every good characterization has the corollary that the decision problem corresponding to the property it characterizes lies in \( \mathrm{{NP}} \cap \) co-NP. ## Infinite sets This appendix gives a minimum-fuss summary of the set-theoretic notions and facts, such as Zorn's lemma and transfinite induction, that are used in Chapter 8. Let \( A, B \) be sets. If there exists a bijective map between \( A \) and \( B \) , we write \( \left| A\right| = \left| B\right| \) and say that \( A \) and \( B \) have the same cardinality. This is clearly an equivalence relation between sets, and we may think of the cardinality \( \left| A\right| \) of \( A \) as the equivalence class containing \( A \) . We write cardinality \( \left| A\right| \leq \left| B\right| \) if there exists an injective map \( A \rightarrow B \) . This is clearly well-defined, and it is a partial ordering: if there are injective maps \( A \rightarrow B \) and \( B \rightarrow A \), there is also a bijection \( A \rightarrow B \cdot {}^{1} \) For every set there exists another that is bigger; for example, \( \left| A\right| < \left| B\right| \) when \( B \) is the power set of \( A \), the set of all its subsets. The natural numbers are defined inductively as \( n \mathrel{\text{:=}} \{ 0,\ldots, n - 1\} \) , \( \mathrm{N} \) starting with \( 0 \mathrel{\text{:=}} \varnothing \) . The usual expression of \( \left| A\right| = n \) can then be read more formally as an abbreviation for \( \left| A\right| = \left| n\right| \) . A set \( A \) is finite if there is a natural number \( n \) such that \( \left| A\right| = n \) ; otherwise it is infinite. \( A \) is countable if \( \left| A\right| \leq \left| \mathbb{N}\right| \), and countably infinite if \( \left| A\right| = \left| \mathbb{N}\right| \) . A bijection \( \mathbb{N} \rightarrow A \) is an enumeration of \( A \) . If \( A \) is infinite then \( \left| \mathbb{N}\right| \leq \left| A\right| \) . Thus, \( \left| \mathbb{N}\right| \) is the smallest infinite cardinality; it is denoted by \( {\aleph }_{0} \) . There is also a smallest uncountable cardinality, denoted by \( {\aleph }_{1} \) . If \( \left| A\right| = \left| \mathbb{R}\right| \) then \( A \) is uncountable, and we say that \( A \) has continuum many elements. For example, there are continuum many infinite 0-1 sequences. (Whether \( \left| \mathbb{R}\right| \) is equal to \( {\aleph }_{1} \) or greater depends on the axioms of set theory assumed; in our context, this question does not arise.) We remark that if \( A \) is infinite and its elements are countable sets, then the union of all these sets is no bigger than \( A \) itself: \( \left| {\bigcup A}\right| = \left| A\right| \) . --- 1 This is the Cantor-Bernstein theorem; a simple graph-theoretic proof is given in Proposition 8.4.6. --- An element \( x \) of a partially ordered set \( X \) is minimal in \( X \) if there is no \( y \in X \) with \( y < x \), and maximal if there is no \( z \in X \) with \( x < z \) . A partially ordered set may have one or many elements that are maximal or minimal, or none at all. An upper bound in \( X \) of a subset \( Y \subseteq X \) is any \( x \in X \) such that \( y \leq x \) for all \( y \in Y \) . A chain is a partially ordered set in which every two elements are comparable. If \( \left( {C, \leq }\right) \) is a chain, and if \( x, y \in C \) satisfy \( x < y \) but no element \( z \) of \( C \) is such that \( x < z < y \), then \( x \) is called the predecessor --- successor --- of \( y \) in \( C \), and \( y \) the successor of \( x \) . A set of the form \( \{ x \in C \mid x < z\} \) , for a given \( z \in C \), is a proper initial segment of \( C \) . A partially ordered set \( \left( {X, \leq }\right) \) is well-founded if every non-empty subset of \( X \) has a minimal element, and a well-founded chain is said well- to be well-ordered. For example, \( \mathbb{N},\mathbb{Z} \)

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x \), and maximal if there is no \( z \in X \) with \( x < z \) . A partially ordered set may have one or many elements that are maximal or minimal, or none at all. An upper bound in \( X \) of a subset \( Y \subseteq X \) is any \( x \in X \) such that \( y \leq x \) for all \( y \in Y \) . A chain is a partially ordered set in which every two elements are comparable. If \( \left( {C, \leq }\right) \) is a chain, and if \( x, y \in C \) satisfy \( x < y \) but no element \( z \) of \( C \) is such that \( x < z < y \), then \( x \) is called the predecessor --- successor --- of \( y \) in \( C \), and \( y \) the successor of \( x \) . A set of the form \( \{ x \in C \mid x < z\} \) , for a given \( z \in C \), is a proper initial segment of \( C \) . A partially ordered set \( \left( {X, \leq }\right) \) is well-founded if every non-empty subset of \( X \) has a minimal element, and a well-founded chain is said well- to be well-ordered. For example, \( \mathbb{N},\mathbb{Z} \) and \( \mathbb{R} \) are all chains (with their ordering usual orderings), but only \( \mathbb{N} \) is well-ordered. Note that every element \( x \) of a well-ordered set \( X \) has a successor (unless \( x \) is maximal in \( X \) ): the unique minimal element of \( \{ y \in X \mid x < y\} \subset X \) . However, an element of a well-ordered set need not have a predecessor, even if it is not minimal. limit An element that has no predecessor is called a limit; for example, the number 1 is a limit in the well-ordered set \[ A = \{ 1 - \frac{1}{n + 1} \mid n \in \mathbb{N}\} \cup \{ 2 - \frac{1}{n + 1} \mid n \in \mathbb{N}\} \] of rationals. One of the many statements equivalent to the axiom of choice (which we assume throughout) is that for every set \( X \) there exists a relation by which \( X \) is well-ordered: ## Well-ordering theorem. Every set can be well-ordered. Two well-ordered sets are said to have the same order type if there is a bijection between them which preserves their orders. Thus \( \mathbb{N} \) and the set of even natural numbers have the same order type, but this differs from the order type of the set \( A \) defined above. Having the same order type is clearly an equivalence relation, which justifies the term if we think of those order types themselves as equivalence classes. When one considers properties shared by all well-ordered sets of the same order type, it is convenient to represent each order type by a --- ordinals --- specially chosen set of that type, its ordinal. The ordinal representing the order type of \( \mathbb{N} \), for instance, is by custom denoted as \( \omega \) ; our example above thus says that the set of even natural numbers has (the) order type (of) \( \omega \) . Finite chains of the same cardinality always have the same order type; we choose \( n \) as the ordinal representing the chains of order \( n \) . If an ordinal \( \beta \) has the same order type as a proper initial segment of another ordinal \( \alpha \), we write \( \beta < \alpha \) . For example, we have \( 0 \leq n < \omega \) for every natural number \( n \) . It can be shown that \( < \) defines an ordering, even a well-ordering, on every set of ordinals. On \( \mathbb{N} \), this ordering coincides with the usual one, so our notation is unambiguous. Since a set \( S \) of ordinals is itself well-ordered, it has an order type-just like any other well-ordered set. If the ordinal \( \alpha \) is a strict upper bound for \( S \), then the order type of \( S \) is at most \( \alpha \) ; it is equal to \( \alpha \) if \( S \) consists of all the ordinals up to (but excluding) \( \alpha \) . In fact, just like the natural numbers, infinite ordinals are usually defined in such a way that \( \alpha \) and \( \{ \beta \mid \beta < \alpha \} \) are actually identical; then our ordering \( < \) for ordinals coincides with the relation \( \in \) . This makes it natural to write a well-ordered set \( S \), of order type \( \alpha \) say, as a family \( S = \left\{ {{s}_{\beta } \mid \beta < \alpha }\right\} \) with \( {s}_{\gamma } < {s}_{\beta } \) for all \( \gamma < \beta < \alpha \) . This is common practice when one proves statements about the elements of --- transfinite induction --- \( S \) by transfinite induction, which works as follows. Suppose we want to show that every \( s \in S \) satisfies some proposition \( P \) ; let us write \( P\left( s\right) \) to express that it does. Just as in ordinary induction we prove, for every \( \beta < \alpha \), that if \( P \) holds for every \( {s}_{\gamma } \) with \( \gamma < \beta \) then \( P \) also holds for \( {s}_{\beta } \) . In practice, we usually have to distinguish the two cases of \( \beta \) being a limit ordinal or a successor. Checking \( P\left( {s}_{0}\right) \) from first principles, as in ordinary induction, is part of the first case, because 0 counts as a limit and the premise of \( {P}_{\gamma } \) for all \( \gamma < 0 \) is void. The conclusion then is that \( P\left( {s}_{\beta }\right) \) for every \( \beta < \alpha \), that is, every \( s \in S \) satisfies \( P \) . This is certainly simple but is it correct? Well, any proper justification of transfinite induction requires a formal treatment of set theory, but so does ordinary induction. Informally, what we have shown is that the set \[ \left\{ {\beta < \alpha \mid P\left( {s}_{\beta }\right) \text{ fails }}\right\} \] has no least element. Since it is well-ordered, it must therefore be empty, so \( P\left( {s}_{\beta }\right) \) holds for all \( \beta < \alpha \) . Similarly, we may define things inductively. Such a recursive defi- --- recursive definition --- nition specifies for each ordinal \( \alpha \) some object \( {x}_{\alpha } \), in a way that may refer to the objects \( {x}_{\beta } \) with \( \beta < \alpha \) (which we think of as ’having been defined earlier'). Our definition of the natural numbers at the start of this appendix is a simple example. In practice, the definition of \( {x}_{\alpha } \) often makes sense only for ordinals \( \alpha \) less than some fixed ordinal \( {\alpha }^{ * } \), although the smallest such \( {\alpha }^{ * } \) may not be known in advance. For example, if the \( {x}_{\alpha } \) are to be distinct vertices picked recursively from a graph \( G \) according to some given rules, it is clear that we shall not be able to find such \( {x}_{\alpha } \) for all \( \alpha < {\alpha }^{ * } \) when \( \left| {\alpha }^{ * }\right| > \left| G\right| \), because \( \alpha \mapsto {x}_{\alpha } \) would be an injective map from \( {\alpha }^{ * } \) to \( V\left( G\right) \) showing that \( \left| {\alpha }^{ * }\right| \leq \left| G\right| \) . Since there exist ordinals larger than \( \left| G\right| \), such as any ordinal equivalent to a well-ordering of the power set of \( V\left( G\right) \), this means that our recursion cannot go on indefinitely, i.e. we shall not be able to define \( {x}_{\alpha } \) for all ordinals \( \alpha \) . We may not know which is the smallest ordinal \( \alpha \) at which the recursion gets stuck, i.e. for which \( {x}_{\alpha } \) cannot be found in compliance with our rules. But this does not matter: we simply define \( {\alpha }^{ * } \) as the first ordinal \( \alpha \) for which \( {x}_{\alpha } \) cannot be found, content ourselves with having defined \( {x}_{\alpha } \) for all \( \alpha < {\alpha }^{ * } \), and say that our recursion terminates at step \( {\alpha }^{ * } \) . (In fact, we usually want a recursive definition to terminate. In our example, we might wish to consider the set of all vertices \( x \in G \) that got picked by our definition, and this will be the set \( \left\{ {{x}_{\alpha } \mid \alpha < {\alpha }^{ * }}\right\} \) .) Finally, our recursive definition for \( {x}_{\alpha } \) may involve choices. In our example, \( {x}_{\alpha } \) might be required to be a neighbour of some \( {x}_{\beta } \) with \( \beta < \alpha \) , but there may be several such \( {x}_{\beta } \), each with several neighbours that have not yet been picked. This does not cause our recursion to get stuck at step \( \alpha \) : we just pick one eligible vertex as \( {x}_{\alpha } \), and proceed. In other words, we accept \( \left\{ {{x}_{\alpha } \mid \alpha < {\alpha }^{ * }}\right\} \) as a properly defined set even though we may not ’know’ its elements \( {x}_{\alpha } \) constructively. Finally, here is a formal statement of Zorn's lemma: Zorn’s Lemma. Let \( \left( {X, \leq }\right) \) be a partially ordered set such that every chain in \( X \) has an upper bound in \( X \) . Then \( X \) contains at least one maximal element. Note that, in applications of Zorn’s lemma, the relation \( \leq \) need not correspond to an intuitive notion of 'smaller than'. Applied to sets or to graphs, for example, it can stand for ’ \( \supseteq \) ’ just as much as for ’ \( \subseteq \) ’. Then the ’upper bound’ of a chain \( \mathcal{C} \) is typically its overall intersection \( \bigcap \mathcal{C} \) . ## Surfaces This appendix offers a summary of background information about surfaces, as needed for an understanding of their role in the proof of the graph minor theorem or the proof of the 'general Kuratowski theorem' for arbitrary surfaces given in Chapter 12.5. In order to be read at a rigorous level it requires familiarity with some basic definitions of general topology (such as of the product and the identification topology), but no more. A surface, for the purpose of this book, is a compact connected \( {}^{1} \) surface Hausdorff topological space \( S \) in which every point has a neighbourhood homeomorphic to the Euclidean plane \( {\mathbb{R}}^{2} \) . An arc, a circle, and a disc arc in \( S \) are subsets that are homeomorphic in the subspace topology to the circle \( {S}^{1} \) real interval \( \left\lbrack {0,1}\right\rbrack \), to the unit circle \( {S}^{1} = \left\{ {x \in {\mat

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chain \( \mathcal{C} \) is typically its overall intersection \( \bigcap \mathcal{C} \) . ## Surfaces This appendix offers a summary of background information about surfaces, as needed for an understanding of their role in the proof of the graph minor theorem or the proof of the 'general Kuratowski theorem' for arbitrary surfaces given in Chapter 12.5. In order to be read at a rigorous level it requires familiarity with some basic definitions of general topology (such as of the product and the identification topology), but no more. A surface, for the purpose of this book, is a compact connected \( {}^{1} \) surface Hausdorff topological space \( S \) in which every point has a neighbourhood homeomorphic to the Euclidean plane \( {\mathbb{R}}^{2} \) . An arc, a circle, and a disc arc in \( S \) are subsets that are homeomorphic in the subspace topology to the circle \( {S}^{1} \) real interval \( \left\lbrack {0,1}\right\rbrack \), to the unit circle \( {S}^{1} = \left\{ {x \in {\mathbb{R}}^{2} : \parallel x\parallel = 1}\right\} \), and to disc the unit disc \( \left\{ {x \in {\mathbb{R}}^{2} : \parallel x\parallel \leq 1}\right\} \) or \( \left\{ {x \in {\mathbb{R}}^{2} : \parallel x\parallel < 1}\right\} \), respectively. The components of a subset \( X \) of \( S \) are the equivalence classes of component points in \( X \) where two points are equivalent if they can be joined by an arc in \( X \) . The surface \( S \) itself, being connected, has only one component. The frontier of \( X \) is the set of all points \( y \) in \( S \) such that every frontier neighbourhood of \( y \) meets both \( X \) and \( S \smallsetminus X \) . The frontier \( F \) of \( X \) separates \( S \smallsetminus X \) from \( X \) : since \( X \cup F \) is closed, every arc from \( S \smallsetminus X \) to \( X \) has a first point in \( X \cup F \), which must lie in \( F \) . A component of the frontier of \( X \) that is a circle in \( S \) is a boundary circle of \( X \) . A boundary boundary circle circle of a disc in \( S \) is said to bound that disc. There is a fundamental theorem about surfaces, their classification. This says that, up to homeomorphism, every surface can be obtained from the sphere \( {S}^{2} = \left\{ {x \in {\mathbb{R}}^{3} : \parallel x\parallel = 1}\right\} \) by ’adding finitely many sphere \( {S}^{2} \) handles or finitely many crosscaps', and that surfaces obtained by adding different numbers of handles or crosscaps are distinct. We shall not need --- 1 Throughout this appendix, 'connected' means 'arc-connected'. --- the classification theorem, but to form a picture \( {}^{2} \) let us see what the handle above operations mean. To add a handle to a surface \( S \), we remove two open discs whose closures in \( S \) are disjoint, and identify \( {}^{3} \) their boundary circles with the circles \( {S}^{1} \times \{ 0\} \) and \( {S}^{1} \times \{ 1\} \) of a copy of \( {S}^{1} \times \left\lbrack {0,1}\right\rbrack \) crosscap disjoint from \( S \) . To add a crosscap, we remove one open disc, and then identify opposite points on its boundary circle in pairs. In order to see that these operations do indeed give new surfaces, we have to check that every identification point ends up with a neighbourhood homeomorphic to \( {\mathbb{R}}^{2} \) . To do this rigorously, let us first look at circles more generally. cylinder A cylinder is the product space \( {S}^{1} \times \left\lbrack {0,1}\right\rbrack \), or any space homeomorphic to it. Its middle circle is the circle \( {S}^{1} \times \left\{ \frac{1}{2}\right\} \) . A Möbius strip Möbius is any space homeomorphic to the product space \( \left\lbrack {0,1}\right\rbrack \times \left\lbrack {0,1}\right\rbrack \) after strip identification of \( \left( {1, y}\right) \) with \( \left( {0,1 - y}\right) \) for all \( y \in \left\lbrack {0,1}\right\rbrack \) . Its middle circle is the set \( \left\{ {\left( {x,\frac{1}{2}}\right) \mid 0 < x < 1}\right\} \cup \{ p\} \), where \( p \) is the point resulting from --- strip neighbourhood --- the identification of \( \left( {1,\frac{1}{2}}\right) \) with \( \left( {0,\frac{1}{2}}\right) \) . It can be shown \( {}^{4} \) that every circle \( C \) in a surface \( S \) is the middle circle of a suitable cylinder or Möbius strip \( N \) in \( S \), which can be chosen small enough to avoid any given compact subset of \( S \smallsetminus C \) . If this strip neighbourhood is a cylinder, then \( N \smallsetminus C \) has two-sided two components and we call \( C \) two-sided; if it is a Möbius strip, then one-sided \( N \smallsetminus C \) has only one component and we call \( C \) one-sided. Using small neighbourhoods inside a strip neighbourhood of the (two-sided) boundary circle of the disc or discs we removed from \( S \) in order to attach a crosscap or handle, one can show easily that both operations do produce new surfaces. Since \( S \) is connected, \( S \smallsetminus C \) cannot have more components than --- separating circle --- \( N \smallsetminus C \) . If \( S \smallsetminus C \) has two components, we call \( C \) a separating circle in \( S \) ; if it has only one, then \( C \) is non-separating. While one-sided circles are obviously non-separating, two-sided circles can be either separating or non-separating. For example, the middle circle of a cylinder added to \( S \) as a ’handle’ is a two-sided non-separating circle in the new surface obtained. When \( {S}^{\prime } \) is obtained from \( S \) by adding a crosscap in place of a disc \( D \), then every arc in \( S \) that runs half-way round the boundary circle of \( D \) becomes a one-sided circle in \( {S}^{\prime } \) . The classification theorem thus has the following corollary: Lemma B.1. Every surface other than the sphere contains a nonseparating circle. --- 2 Compare also Figure B.1. 3 This is made precise by the identification topology, whose formal definition can be found in any topology book. 4 In principle, the strip neighbourhood \( N \) is constructed as in the proof of Lemma 4.2.2, using the compactness of \( C \) . However since we are not in a piecewise linear setting now, the construction is considerably more complicated. --- We shall see below that, in a sense, our two examples of non-separating circles are all there are: cutting a surface along any non-separating circle (and patching up the holes) will always produce a surface with fewer handles or crosscaps. An embedding \( G \hookrightarrow S \) of a graph \( G \) in \( S \) is a map \( \sigma \) that maps the --- embedding \( \sigma : G \hookrightarrow S \) --- vertices of \( G \) to distinct points in \( S \) and its edges \( {xy} \) to \( \sigma \left( x\right) - \sigma \left( y\right) \) arcs in \( S \), so that no inner point of such an arc is the image of a vertex or lies on another arc. We then write \( \sigma \left( G\right) \) for the union of all those points and arcs in \( S \) . A face of \( G \) in \( S \) is a component of \( S \smallsetminus \sigma \left( G\right) \), and the face subgraph of \( G \) that \( \sigma \) maps to the frontier of this face is its boundary. boundary Note that while faces in the sphere are always discs (if \( G \) is connected), in general they need not be. One can prove that in every surface one can embed a suitable graph so that every face becomes a disc. The following general version of Euler's theorem 4.2.9 therefore applies to all surfaces: Theorem B.2. For every surface \( S \) there exists an integer \( \chi \left( S\right) \) such that whenever a graph \( G \) with \( n \) vertices and \( m \) edges is embedded in \( S \) so that there are \( \ell \) faces and every face is a disc, we have \[ n - m + \ell = \chi \left( S\right) \] This invariant \( \chi \) of \( S \) is its Euler characteristic. For computational simplicity we usually work instead with the derived invariant \[ \varepsilon \left( S\right) \mathrel{\text{:=}} 2 - \chi \left( S\right) \] \( \varepsilon \left( S\right) \) the Euler genus of \( S \), because \( \chi \) is negative for most surfaces but \( \varepsilon \) takes Euler genus its values in \( \mathbb{N} \) (see below). Perhaps the most striking feature of Euler's theorem is that it works with almost any graph embedded in \( S \) . This makes it easy to see how the Euler genus is affected by the addition of a handle or crosscap. Indeed, let \( D \) and \( {D}^{\prime } \) be two open discs in \( S \) that we wish to remove in order to attach a handle there. Let \( G \) be any graph embedded in \( S \) so that every face is a disc. If necessary, shift \( G \) on \( S \) so that \( D \) and \( {D}^{\prime } \) each lie inside a face, \( f \) and \( {f}^{\prime } \), say. Add cycles \( C \) and \( {C}^{\prime } \) on the boundary circles of \( D \) and \( {D}^{\prime } \), and join them by an edge to the old boundaries of \( f \) and \( {f}^{\prime } \), respectively. Then every face of the resulting graph is again a disc, and \( D \) and \( {D}^{\prime } \) are among these. Now remove \( D \) and \( {D}^{\prime } \), and add a handle with an additional \( C - {C}^{\prime } \) edge running along it. This operation makes the new handle into one new face, which is a disc. It thus reduces the total number of faces by 1 (since we lost \( D \) and \( {D}^{\prime } \) but gained the new face on the handle) and increases the number of edges by 1 , but leaves the number of vertices unchanged. As a result, \( \varepsilon \) grows by 2 . Similarly, replacing a disc \( D \) bounded by a cycle \( C \subseteq G \) with a crosscap decreases the number of faces by 1 (since we lose \( D \) ), but leaves \( n - m \) unchanged if we arrange the cycle \( C \) in such a way that vertices get identified with vertices when we identify opposite points. We have thus shown the following: ## Lemma B.3. (i) Adding a handle to a surface raises its Euler genus by 2. (ii) Ad

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hen every face of the resulting graph is again a disc, and \( D \) and \( {D}^{\prime } \) are among these. Now remove \( D \) and \( {D}^{\prime } \), and add a handle with an additional \( C - {C}^{\prime } \) edge running along it. This operation makes the new handle into one new face, which is a disc. It thus reduces the total number of faces by 1 (since we lost \( D \) and \( {D}^{\prime } \) but gained the new face on the handle) and increases the number of edges by 1 , but leaves the number of vertices unchanged. As a result, \( \varepsilon \) grows by 2 . Similarly, replacing a disc \( D \) bounded by a cycle \( C \subseteq G \) with a crosscap decreases the number of faces by 1 (since we lose \( D \) ), but leaves \( n - m \) unchanged if we arrange the cycle \( C \) in such a way that vertices get identified with vertices when we identify opposite points. We have thus shown the following: ## Lemma B.3. (i) Adding a handle to a surface raises its Euler genus by 2. (ii) Adding a crosscap to a surface raises its Euler genus by 1. Since the sphere has Euler genus 0 (Theorem 4.2.9), the classification theorem and Lemma B. 3 tell us that \( \varepsilon \) has all its values in \( \mathbb{N} \) . We may thus try to prove theorems about surfaces by induction on \( \varepsilon \) . For the induction step, we could simply undo the addition of a handle or crosscap described earlier, cutting along the new non-separating circle it produced (which runs around the new handle or 'half-way' around the crosscap) and restoring the old surface by putting back the disc or discs we removed. A problem with this is that we do not normally know where on our surface this circle lies, say with respect to a given graph embedded in it. However, the genus-reducing cut-and-paste operation can be carried out with any non-separating circle: we do not have to use one that we know came from a new handle or crosscap. This is an example of a more general technique known as surgery, and works as follows. Let \( C \) be a non-separating circle in a surface \( S \neq {S}^{2} \) . To cut \( S \) cutting along \( C \), we form a new space \( {S}^{\prime } \) from \( S \) by replacing every point \( x \in C \) with two points \( {x}^{\prime },{x}^{\prime \prime } \) and defining the topology on the modified set as follows. \( {}^{5} \) Let \( N \) be any strip neighbourhood of \( C \) in \( S \), and put \( {X}^{\prime } \mathrel{\text{:=}} \) \( \left\{ {{x}^{\prime } \mid x \in C}\right\} \) and \( {X}^{\prime \prime } \mathrel{\text{:=}} \left\{ {{x}^{\prime \prime } \mid x \in C}\right\} \) . If \( N \) is a cylinder, then \( N \smallsetminus C \) has two components \( {N}^{\prime } \) and \( {N}^{\prime \prime } \), and we choose the neighbourhoods of the new points \( {x}^{\prime } \) and \( {x}^{\prime \prime } \) in \( {S}^{\prime } \) so that \( {X}^{\prime } \) and \( {X}^{\prime \prime } \) become boundary circles of \( {N}^{\prime } \) and \( {N}^{\prime \prime } \) in \( {S}^{\prime } \), respectively, and \( {N}^{\prime } \cup {X}^{\prime } \) and \( {N}^{\prime \prime } \cup {X}^{\prime \prime } \) become disjoint cylinders in \( {S}^{\prime } \) . If \( N \) is a Möbius strip, we choose these neighbourhoods so that \( {X}^{\prime } \) and \( {X}^{\prime \prime } \) each form an arc in \( {S}^{\prime } \) and \( {X}^{\prime } \cup {X}^{\prime \prime } \) is a boundary circle of \( N \smallsetminus C \) in \( {S}^{\prime } \), with \( \left( {N \smallsetminus C}\right) \cup {X}^{\prime } \cup {X}^{\prime \prime } \) forming one cylinder in \( {S}^{\prime } \) . --- capping --- Finally, we turn \( {S}^{\prime } \) into a surface by capping its holes: for each of the (two or one) boundary circles \( {X}^{\prime } \) and \( {X}^{\prime \prime } \) or \( {X}^{\prime } \cup {X}^{\prime \prime } \) of \( S \smallsetminus C \) in \( {S}^{\prime } \) we take a disc disjoint from \( {S}^{\prime } \) and identify its boundary circle with \( {X}^{\prime },{X}^{\prime \prime } \) or \( {X}^{\prime } \cup {X}^{\prime \prime } \), respectively, so that the space obtained is again a surface. --- 5 The description that follows may sound complicated, but it is not: working in our concrete models of the cylinder and the Möbius strip it is easy to write down an explicit neighbourhood basis that defines a topology with the properties stated. As all we want is to obtain some surface of smaller genus, we do not care about uniqueness (which will follow anyhow from Lemma B. 4 and the classification theorem). --- Computing how these operations affect the Euler genus of \( S \) is again easy, assuming we can embed a graph in \( S \) so that every face is a disc and \( C \) is the image of a cycle. (This can always be done, but it is not easy to prove. \( {}^{6} \) ) Indeed, by doubling \( C \) we left \( n - m \) unchanged, because a cycle has the same number of vertices as edges. So all we changed was \( \ell \) , which increased by 2 in the first case and by 1 in the second. Lemma B.4. Let \( C \) be any non-separating circle in a surface \( S \), and let \( {S}^{\prime } \) be obtained from \( S \) by cutting along \( C \) and capping the hole or holes. (i) If \( C \) is one-sided in \( S \), then \( \varepsilon \left( {S}^{\prime }\right) = \varepsilon \left( S\right) - 1 \) . (ii) If \( C \) is two-sided in \( S \), then \( \varepsilon \left( {S}^{\prime }\right) = \varepsilon \left( S\right) - 2 \) . Lemma B. 4 gives us a large supply of circles to cut along in an induction on the Euler genus. Still, it is sometimes more convenient to cut along a separating circle, and many of these can be used too: Lemma B.5. Let \( C \) be a separating circle in a surface \( S \), and let \( {S}^{\prime } \) and \( {S}^{\prime \prime } \) be the two surfaces obtained from \( S \) by cutting along \( C \) and capping the holes. Then \[ \varepsilon \left( S\right) = \varepsilon \left( {S}^{\prime }\right) + \varepsilon \left( {S}^{\prime \prime }\right) . \] In particular, if \( C \) does not bound a disc in \( S \), both \( {S}^{\prime } \) and \( {S}^{\prime \prime } \) have smaller Euler genus than \( S \) . Proof. As before, embed a graph \( G \) in \( S \) so that every face is a disc and \( C \) is the image of a cycle in \( G \), and let \( {G}^{\prime } \hookrightarrow {S}^{\prime } \) and \( {G}^{\prime \prime } \hookrightarrow {S}^{\prime \prime } \) be the two graphs obtained in the surgery. Thus, \( {G}^{\prime } \) and \( {G}^{\prime \prime } \) both contain a copy of the cycle on \( C \), which we assume to have \( k \) vertices and edges. Then, with the obvious notation, we have \[ \varepsilon \left( {S}^{\prime }\right) + \varepsilon \left( {S}^{\prime \prime }\right) = \left( {2 - {n}^{\prime } + {m}^{\prime } - {\ell }^{\prime }}\right) + \left( {2 - {n}^{\prime \prime } + {m}^{\prime \prime } - {\ell }^{\prime \prime }}\right) \] \[ = 4 - \left( {n + k}\right) + \left( {m + k}\right) - \left( {\ell + 2}\right) \] \[ = 2 - n + m - \ell \] \[ = \varepsilon \left( S\right) \text{.} \] Now if \( {S}^{\prime } \) (say) is a sphere, then \( {S}^{\prime } \cap S \) was a disc in \( S \) bounded by \( C \) . Hence, if \( C \) does not bound a disc in \( S \) then \( \varepsilon \left( {S}^{\prime }\right) \) and \( \varepsilon \left( {S}^{\prime \prime }\right) \) are both non-zero, giving the second statement of the lemma. We now apply these techniques to prove a lemma for our direct proof in Chapter 12 of the 'Kuratowski theorem for arbitrary surfaces', Corollary 12.5.3. 6 Perhaps the simplest proof was given by C. Thomassen, The Jordan-Schoenflies theorem and the classification of surfaces, Amer. Math. Monthly 99 (1992), 116-130. \( \left\lbrack {12.5.4}\right\rbrack \) Lemma B.6. Let \( S \) be a surface, and let \( \mathcal{C} \) be a finite set of disjoint circles in \( S \) . Assume that none of these circles bounds a disc in \( S \), and that \( S \smallsetminus \bigcup \mathcal{C} \) has a component \( {D}_{0} \) whose closure in \( S \) meets every circle in \( \mathcal{C} \) . Then \( \varepsilon \left( S\right) \geq \left| \mathcal{C}\right| \) . Proof. We begin with the observation that the closure of \( {D}_{0} \) not only meets but even contains every circle \( C \in \mathcal{C} \) . This is because \( C \) has a strip neighbourhood \( N \) disjoint from all the other circles in \( \mathcal{C} \) (since their union is compact), and each of the (one or two) components of \( N \smallsetminus C \) has all of \( C \) in its closure. Since \( {D}_{0} \) meets, and hence contains, at least one component of \( N \smallsetminus C \), its closure contains \( C \) . \( {\mathcal{C}}_{1},{\mathcal{C}}_{2}^{1},{\mathcal{C}}_{2}^{2} \) Let us partition \( \mathcal{C} \) as \( \mathcal{C} = {\mathcal{C}}_{1} \cup {\mathcal{C}}_{2}^{1} \cup {\mathcal{C}}_{2}^{2} \), where the circles in \( {\mathcal{C}}_{1} \) are one-sided, those in \( {\mathcal{C}}_{2}^{1} \) are two-sided but non-separating, and those in \( {\mathcal{C}}_{2}^{2} \) are separating. We shall, in turn, cut along all the circles in \( {\mathcal{C}}_{1} \) , some \( \left| {\mathcal{C}}_{2}^{2}\right| \) circles not in \( \mathcal{C} \), and at least half the circles in \( {\mathcal{C}}_{2}^{1} \) . This will \( {S}_{0},\ldots ,{S}_{n} \) give us a sequence \( {S}_{0},\ldots ,{S}_{n} \) of surfaces, where \( {S}_{0} = S \), and \( {S}_{i + 1} \) is \( {C}_{i} \) obtained from \( {S}_{i} \) by cutting along a circle \( {C}_{i} \) and capping the hole(s). Our task will be to ensure that \( {C}_{i} \) is non-separating in \( {S}_{i} \) for every \( i = 0,\ldots, n - 1 \) . Then Lemma B. 4 will imply that \( \varepsilon \left( {S}_{i + 1}\right) \leq \varepsilon \left( {S}_{i}\right) - 1 \) for all \( i \) and \( \varepsilon \left( {S}_{i + 1}\right) \leq \varepsilon \left( {S}_{i}\right) - 2 \) whenever \( {C}_{i} \in {\mathcal{C}}_{2}^{1} \), giving \[

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( {\mathcal{C}}_{2}^{1} \) are two-sided but non-separating, and those in \( {\mathcal{C}}_{2}^{2} \) are separating. We shall, in turn, cut along all the circles in \( {\mathcal{C}}_{1} \) , some \( \left| {\mathcal{C}}_{2}^{2}\right| \) circles not in \( \mathcal{C} \), and at least half the circles in \( {\mathcal{C}}_{2}^{1} \) . This will \( {S}_{0},\ldots ,{S}_{n} \) give us a sequence \( {S}_{0},\ldots ,{S}_{n} \) of surfaces, where \( {S}_{0} = S \), and \( {S}_{i + 1} \) is \( {C}_{i} \) obtained from \( {S}_{i} \) by cutting along a circle \( {C}_{i} \) and capping the hole(s). Our task will be to ensure that \( {C}_{i} \) is non-separating in \( {S}_{i} \) for every \( i = 0,\ldots, n - 1 \) . Then Lemma B. 4 will imply that \( \varepsilon \left( {S}_{i + 1}\right) \leq \varepsilon \left( {S}_{i}\right) - 1 \) for all \( i \) and \( \varepsilon \left( {S}_{i + 1}\right) \leq \varepsilon \left( {S}_{i}\right) - 2 \) whenever \( {C}_{i} \in {\mathcal{C}}_{2}^{1} \), giving \[ \varepsilon \left( S\right) \geq \varepsilon \left( {S}_{n}\right) + \left| {\mathcal{C}}_{1}\right| + \left| {\mathcal{C}}_{2}^{2}\right| + 2\left| {\mathcal{C}}_{2}^{1}\right| /2 \geq \left| \mathcal{C}\right| \] as desired.  Fig. B.1. Cutting the 1-sided circle \( {C}_{1} \) and the 2-sided circles \( {C}_{2},{C}_{3} \) and \( {C}_{5},{C}_{7},{C}_{8} \) and \( {C}_{9}^{\prime } \) does not separate \( S \) Cutting along the circles in \( {\mathcal{C}}_{1} \) (and capping the holes) is straightforward: since these circles are one-sided, they are always non-separating. Next, we consider the circles in \( {\mathcal{C}}_{2}^{2} \), such as \( {C}_{9} \) in Figure B.1. For every \( C \in {\mathcal{C}}_{2}^{2} \), denote by \( D\left( C\right) \) the component of \( S \smallsetminus C \) that does not contain \( {D}_{0} \) . Since every circle in \( \mathcal{C} \) lies in the closure of \( {D}_{0} \) but no point of \( D\left( C\right) \) does, these \( D\left( C\right) \) are also components of \( S \smallsetminus \bigcup \mathcal{C} \) . In particular, they are disjoint for different \( C \) . Thus, each \( D\left( C\right) \) will also be a component of \( {S}_{i} \smallsetminus C \), where \( {S}_{i} \) is the current surface after any surgery performed on the circles in \( {\mathcal{C}}_{1} \) and inside \( D\left( {C}^{\prime }\right) \) for some \( {C}^{\prime } \neq C \) . Given a fixed circle \( C \in {\mathcal{C}}_{2}^{2} \), let \( {S}^{\prime } \) be the surface obtained from \( D\left( C\right) \) by capping its hole. Since \( C \) does not bound a disc in \( S \), we know that \( {S}^{\prime } \) is not a sphere and hence contains a non-separating circle \( {C}^{\prime } \) (Lemma B.1). We choose \( {C}^{\prime } \) so that it avoids the cap we added to form \( {S}^{\prime } \), i.e. so that \( {C}^{\prime } \subseteq S \smallsetminus C \) . Then \( {C}^{\prime } \) is also non-separating in the current surface \( {S}_{i} \) (since every point of \( {S}_{i} \smallsetminus {C}^{\prime } \) can be joined by an arc in \( {S}_{i} \smallsetminus {C}^{\prime } \) to \( C \) , which is connected), and we may select \( {C}^{\prime } \) as a circle \( {C}_{i} \) to cut along. It remains to select at least half of the circles in \( {\mathcal{C}}_{2}^{1} \) as circles \( {C}_{i} \) to cut along. We begin by selecting all those whose entire strip neighbourhoods (i.e., both their ’sides’) lie in \( {D}_{0} \) . (In Figure B.1, these are the circles \( {C}_{2} \) and \( {C}_{3} \) .) These circles \( C \) are non-separating also in the surface \( {S}_{i} \) current before they are cut, because \( {D}_{0} \) will lie inside a component of \( {S}_{i} \smallsetminus C \) . Every other \( C \in {\mathcal{C}}_{2}^{1} \) lies in the closure also of a component \( D\left( C\right) \neq {D}_{0} \) of \( S \smallsetminus \bigcup \mathcal{C} \) . (In Figure B.1, these are the circles \( {C}_{4},\ldots ,{C}_{8} \) .) For every component \( D \) of \( S \smallsetminus \bigcup \mathcal{C} \) we select all but one of the circles \( C \in {\mathcal{C}}_{2}^{1} \) with \( D\left( C\right) = D \) as a cutting circle \( {C}_{i} \) . Clearly, each of these \( {C}_{i} \) will be non-separating also in its current surface \( {S}_{i} \), and their total number at least \( \left| {\mathcal{C}}_{2}^{1}\right| /2 \) . # Hints for all the Exercises Caveat. These hints are intended to set on the right track anyone who has already spent some time over an exercise but somehow failed to make much progress. They are not designed to be particularly intelligible without such an initial attempt, and they will rarely spoil the fun by giving away the key idea. They may, however, narrow one's mind by focusing on what is just one of several possible ways to think about a problem... ## Hints for Chapter 1 1. \( {}^{ - } \) How many edges are there at each vertex? 2. Average degree and edges: consider the vertex degrees. Diameter: how do you determine the distance between two vertices from the corresponding 0-1 sequences? Girth: does the graph have a cycle of length 3 ? Circumference: partition the \( d \) -dimensional cube into cubes of lower dimension and use induction. 3. Consider how the path intersects \( C \) . Where can you see cycles, and can they all be short? 4. \( {}^{ - } \) Can you find graphs for which Proposition 1.3.2 holds with equality? 5. Estimate the distances within \( G \) as seen from a central vertex. 6. Count vertices as in the proof of Proposition 1.3.3. For the even case, start with two adjacent vertices. 7. \( {}^{ + } \) Consider a longest path \( P \) in \( G \) . Where do its endvertices have their neighbours? Can \( G\left\lbrack P\right\rbrack \) contain a cycle on \( V\left( P\right) \) ? 8. \( {}^{ + } \) Pick two vertices \( x, y \) of maximum distance, and show that many of the distance classes \( {D}_{i} \) from \( x \) have to be large. 9. \( {}^{ - } \) Assume the contrary, and derive a contradiction. 10. \( {}^{ - } \) Find two vertices that are linked by two independent paths. 11. For each type of graph, the solution requires separate proofs of (coinciding) upper and a lower bounds. For the cube, use induction on \( n \) . 12. \( {}^{ - } \) Try to find counterexamples for \( k = 1 \) . 13. \( {}^{ + } \) Rephrase (i) and (ii) as statements about the existence of two \( \mathbb{N} \rightarrow \mathbb{N} \) functions. To show the equivalence, express each of these functions in terms of the other. Show that (iii) may hold even if (i) and (ii) do not, and strengthen (iii) to remedy this. 14. \( {}^{ + } \) Try to imitate the proof assuming \( \varepsilon \left( G\right) \geq {c}_{k} \) instead of condition (ii). Why does this fail, and why does condition (ii) remedy the problem? 15. Show (i) \( \Rightarrow \) (ii) \( \Rightarrow \) (iii) \( \Rightarrow \) (iv) \( \Rightarrow \) (i) from the definitions of the relevant concepts. 16. \( {}^{ - } \) How can we turn distinct neighbours into distinct leaves? 17. Average degree. 18. Theorem 1.5.1. 19. \( {}^{ + } \) Induction. 20. The easiest solution is to apply induction on \( \left| T\right| \) . What kind of vertex of \( T \) will be best to delete in the induction step? 21. Induction on \( \left| T\right| \) is a possibility, but not the only one. 22. \( {}^{ - } \) Count the edges. 23. Show that if a graph contains any odd cycle at all it also contains an induced one. 24. \( {}^{ + } \) Given a graph \( G \), how would you split its vertex set into two parts \( A \) and \( B \) so that the bipartite graph \( H \) defined by the \( A - B \) edges of \( G \) has minimum degree as large as possible? To find \( f \), apply this method to a suitable subgraph \( G \) of a given graph \( {G}^{\prime } \), and determine how large \( d\left( {G}^{\prime }\right) \) must be to ensure that \( \delta \left( H\right) \geq k \) . 25. Try to carry the proof for finite graphs over to the infinite case. Where does it fail? 26. Try to imitate the proof of Theorem 1.8.1. 27. Why do all the cuts \( E\left( v\right) \) generate the cut space? Will they still do so if we omit one of them? Or even two? 28. Be clear about what exactly the word 'minimal' refers to in its various contexts. 29. Start with the case that the graph considered is a cycle. 30. \( {}^{ + } \) Consider a set \( F \subseteq E \) that meets every cycle in an even number of edges. Contract all edges not in \( F \) . What can you say about the structure of the arising multigraph? 31. Given a cycle \( C \) to be generated, for which edges \( e \) should \( {C}_{e} \) be among the generators of \( C \) ? 32. Given a cut \( D \) to be generated, for which edges \( e \) should \( {D}_{e} \) be among the generators of \( D \) ? 33. Apply Theorem 1.9.6. 34. Induction on \( k \) . 35. \( {}^{ + } \) Apply induction on \( \left| G\right| \) . Delete a vertex \( v \) of odd degree, and apply the induction hypothesis to a suitable modification of \( G - v \) . ## Hints for Chapter 2 1. Recall how an augmenting path turns a given matching into a larger one. Can you reverse this process to obtain an augmenting path from the two matchings? 2. Augmenting paths. 3. Turn the functions into a graph, and consider its components. 4. If there is no matching of \( A \), then by König’s theorem few vertices cover all the edges. How can this assumption help you to find a large subset of \( A \) with few neighbours? 5. Show that the marriage condition fails in \( H \) for \( {A}_{1} \cup {A}_{2} \) . The proof is almost a mirror image of the third proof, with unions and intersections interchanged. 6. \( {}^{ + } \) If you have \( S \subsetneqq {S}^{\prime } \subseteq A \) with \( \left| S\right| = \left| {N\left( S\right) }\right| \) in the finite case, the marriage condition ensures that \( N\left( S\right

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

n \( \left| G\right| \) . Delete a vertex \( v \) of odd degree, and apply the induction hypothesis to a suitable modification of \( G - v \) . ## Hints for Chapter 2 1. Recall how an augmenting path turns a given matching into a larger one. Can you reverse this process to obtain an augmenting path from the two matchings? 2. Augmenting paths. 3. Turn the functions into a graph, and consider its components. 4. If there is no matching of \( A \), then by König’s theorem few vertices cover all the edges. How can this assumption help you to find a large subset of \( A \) with few neighbours? 5. Show that the marriage condition fails in \( H \) for \( {A}_{1} \cup {A}_{2} \) . The proof is almost a mirror image of the third proof, with unions and intersections interchanged. 6. \( {}^{ + } \) If you have \( S \subsetneqq {S}^{\prime } \subseteq A \) with \( \left| S\right| = \left| {N\left( S\right) }\right| \) in the finite case, the marriage condition ensures that \( N\left( S\right) \subsetneqq N\left( {S}^{\prime }\right) \) : increasing \( S \) makes more neighbours available. Use the fact that this fails when \( S \) is infinite. 7. Apply the marriage theorem. 8. Construct a bipartite graph in which \( A \) is one side, and the other side consists of a suitable number of copies of the sets \( {A}_{i} \) . Define the edge set of the graph so that the desired result can be derived from the marriage theorem. 9. \( {}^{ + } \) Construct chains in the power set lattice of \( X \) as follows. For each \( k < n/2 \), use the marriage theorem to find a \( 1 - 1 \) map \( \varphi \) from the set \( A \) of \( k \) -subsets to the set \( B \) of \( \left( {k + 1}\right) \) -subsets of \( X \) such that \( Y \subseteq \varphi \left( Y\right) \) for all \( Y \in A \) . 10. \( {}^{ - } \) Try \( {C}^{6} \) . 11. \( {}^{ - } \) Change occurs most likely if unhappy vertices can bring it about without having to ask the happy ones. (If philosophy does not help, try \( {K}^{3} \) .) 12. Alternating paths. 13. Decide where the leaves should go: in factor-critical components or in \( S \) ? 14. By transitivity, every vertex lies in a set \( S \) as in Theorem 2.2.3. 15. For the 'if' direction apply Tutte's 1-factor theorem to the graph \( G * {K}^{\left| G\right| - {2k}} \), or use the remarks on maximum-cardinality matchings following Theorem 2.2.3. 16. \( {}^{ - } \) Corollary 2.2.2. 17. Let \( G \) be a bipartite graph that satisfies the marriage condition, with bipartition \( \{ A, B\} \) say. Reduce the problem to the case of \( \left| A\right| = \left| B\right| \) . To verify the premise of Tutte’s theorem for a given set \( S \subseteq V\left( G\right) \) , bound \( \left| S\right| \) from below in terms of the number of components of \( G - S \) that contain more vertices from \( A \) than from \( B \) and vice versa. 18. \( {}^{ - } \) For the first task, consider a typical non-bipartite graph. For the second, start with any maximal set of independent edges. 19. Where in the proof of Lemma 2.3.1 do we use that \( \Delta \left( G\right) \leq 3 \) ? 20. Find a subgraph \( H \) isomorphic to a cycle or \( {K}^{2} \) or \( {K}^{1} \) that contains a vertex not adjacent to any vertex in \( G - H \) . Then apply induction on \( \alpha \) . 21. If you cannot spot the error just by reading the proof very carefully (which you should be able to do, really-but this case it is tricky), it is a good idea to test the assertion for extreme cases or small graphs. When you have found a counterexample, go through the proof with this graph in mind and see where exactly it fails. 22. \( {}^{ - } \) Consider any smallest path cover. 23. Direct all the edges from \( A \) to \( B \) . 24. \( {}^{ - } \) Dilworth. 25. Start with the set of minimal elements in \( P \) . 26. Think of the elements of \( A \) as being smaller than their neighbours in \( B \) . 27. Construct a poset from arbitrarily large finite antichains. ## Hints for Chapter 3 1. \( {}^{ - } \) Recall the definitions of ’separate’ and ’component’. 2. Describe in words what the picture suggests. 3. Use Exercise 1 to answer the first question. The second requires an elementary calculation, which the figure may already suggest. 4. Only the first part needs arguing; the second then follows by symmetry. Suppose a component of \( G - X \) is not met by \( {X}^{\prime } \) . Where does \( {X}^{\prime } \) lie in this picture? Remember Exercise 1. 5. \( {}^{ - } \) How can a block fail to be a maximal 2-connected subgraph? And what else follows then? 6. Deduce the connectedness of the block graph from that of the graph itself, and its acyclicity from the maximality of each block. 7. Prove the statement inductively using Proposition 3.1.3. Alternatively, choose a cycle through one of the two vertices and with minimum distance from the other vertex. Show that this distance cannot be positive. 8. Belonging to the same block is an equivalence relation on the edge set; see Exercise 5. 9. Induction along Proposition 3.1.3. 10. Assuming that \( G/{xy} \) is not 3-connected, distinguish the cases when \( {v}_{xy} \) lies inside or outside a separator of at most 2 vertices. 11. \( {\left( \mathrm{i}\right) }^{ - } \) Consider the edges incident with a smaller separator. (ii) Induction shows that all the graphs obtained by the construction are cubic and 3-connected. For the converse, consider a maximal subgraph \( {TH} \subseteq G \) such that \( H \) is constructible as stated; then show that \( H = G \) . 12. \( {}^{ + } \) If such a finite set exists, then every other 3-connected graph can be made into a smaller 3-connected graph by deleting one vertex and suppressing any arising vertices of degree 2. (Why?) For which graphs is this possible? 13. Check the induction. 14. How big is \( S \) ? To recognize the easy remaining case, it helps to have solved the previous exercise first. 15. Choose the disjoint \( A - B \) paths in \( L\left( G\right) \) minimal. 16. Consider a longest cycle \( C \) . How are the other vertices joined to \( C \) ? 17. Consider a cycle through as many of the \( k \) given vertices as possible. If one them is missed, can you re-route the cycle through it? 18. Consider the graph of the hint. Show that any subset of its vertices that meets all \( H \) -paths (but not \( H \) ) corresponds to a similar subset of \( E\left( G\right) \smallsetminus E\left( H\right) \) . What does a pair of independent \( H \) -paths in the auxiliary graph correspond to in \( G \) ? 19. \( {}^{ - } \) How many paths can a single \( {K}^{{2m} + 1} \) accomodate? 20. Choose suitable degrees for the vertices in \( B \) . 21. \( {}^{ + } \) Let \( H \) be the (edgeless) graph on the new vertices. Consider the sets \( X \) and \( F \) that Mader’s theorem provides if \( {G}^{\prime } \) does not contain \( \left| G\right| /2 \) independent \( H \) -paths. If \( G \) has no 1 -factor, use these to find a suitable set that can play the role of \( S \) in Tutte’s theorem. 22. \( {}^{ - } \) If two vertices \( s, t \) are separated by fewer than \( {2k} - 1 \) vertices, extend \( \{ s\} \) and \( \{ t\} \) to \( k \) -sets \( S \) and \( T \) showing that \( G \) is not \( k \) -linked. 23. To construct a highly connected graph that is not \( k \) -linked, start by writing down the vertices \( {s}_{1},\ldots ,{s}_{k},{t}_{1},\ldots ,{t}_{k} \) . By specifying suitable non-edges, make the paths in the required linkage need more vertices in total than there are vertices left in the graph. To make the graph highly connected, add all edges other than the specified non-edges. 24. Use induction on \( {2k} - \left| {S \cup T}\right| \), where \( S \mathrel{\text{:=}} \left\{ {{s}_{1},\ldots ,{s}_{k}}\right\} \) and \( T \mathrel{\text{:=}} \) \( \left\{ {{t}_{1},\ldots ,{t}_{k}}\right\} \) . For the induction step recall that \( \delta \left( G\right) \geq {2k} - 1 \), by Exercise 22. 25. To construct the \( T{K}^{r} \), start by picking the branch vertices and their neighbours. ## Hints for Chapter 4 1. Embed the vertices inductively. Where should you not put the new vertex? 2. Figure 1.6.2. 3. \( {}^{ - } \) Make the given graph connected. 4. This is a generalization of Corollary 4.2.10. 5. Theorem 2.4.4. 6. Imitate the proof of Corollary 4.2.10. 7. Proposition 4.2.7. 8. \( {}^{ - } \) Express the difference between the two drawings as a formal statement about vertices, faces, and the incidences between them. 9. Combinatorially: use the definition. Topologically: express the relative position of the short (respectively, the long) branches of \( {G}^{\prime } \) formally as a property of \( {G}^{\prime } \) which any topological isomorphism would preserve but \( G \) lacks. 10. \( {}^{ - } \) Reflexivity, symmetry, transitivity. 11. Look for a graph whose drawings all look the same, but which admits an automorphism that does not extend to a homeomorphism of the plane. Interpret this automorphism as \( {\sigma }_{2} \circ {\sigma }_{1}^{-1} \) . 12. \( {}^{ + } \) Star-shape: every inner face contains a point that sees the entire face boundary. 13. Work with plane rather than planar graphs. 14. (i) The set \( \mathcal{X} \) may be infinite. (ii) If \( Y \) is a \( {TX} \), then every \( {TY} \) is also a \( {TX} \) . 15. \( {}^{ - } \) By the next exercise, any counterexample can be disconnected by at most two vertices. 16. Incorporate the extra condition into the induction hypothesis of the proof. It may help to disallow polygons with 180 degree angles. 17. Number of edges. 18. Use that maximal planar graphs are 3-connected, and that the neighbours of each vertex induce a cycle. 19. If \( G = {G}_{1} \cup {G}_{2} \) with \( {G}_{1} \cap {G}_{2} = \overline{{K}^{2}} \), we have a problem. This will go away if we embed a little more than necessary. 20. Use a suitable modification of the given graph \( G \) to simulate outerpla-narit

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

e plane. Interpret this automorphism as \( {\sigma }_{2} \circ {\sigma }_{1}^{-1} \) . 12. \( {}^{ + } \) Star-shape: every inner face contains a point that sees the entire face boundary. 13. Work with plane rather than planar graphs. 14. (i) The set \( \mathcal{X} \) may be infinite. (ii) If \( Y \) is a \( {TX} \), then every \( {TY} \) is also a \( {TX} \) . 15. \( {}^{ - } \) By the next exercise, any counterexample can be disconnected by at most two vertices. 16. Incorporate the extra condition into the induction hypothesis of the proof. It may help to disallow polygons with 180 degree angles. 17. Number of edges. 18. Use that maximal planar graphs are 3-connected, and that the neighbours of each vertex induce a cycle. 19. If \( G = {G}_{1} \cup {G}_{2} \) with \( {G}_{1} \cap {G}_{2} = \overline{{K}^{2}} \), we have a problem. This will go away if we embed a little more than necessary. 20. Use a suitable modification of the given graph \( G \) to simulate outerpla-narity. 21. Use the fact that \( \mathcal{C}\left( G\right) \) is the direct sum of \( \mathcal{C}\left( {G}_{1}\right) \) and \( \mathcal{C}\left( {G}_{2}\right) \) . 22.* Euler. 23. The face boundaries generate \( \mathcal{C}\left( G\right) \) . 24. \( {}^{ + } \) Solve the previous exercise first. 25. \( {}^{ - } \) How many vertices does it have? 26. \( {}^{ - } \) Join two given vertices of the dual by a straight line, and use this to find a path between them in the dual graph. 27. \( {}^{ + } \) Define the required bijections \( F \rightarrow {V}^{ * }, E \rightarrow {E}^{ * }, V \rightarrow {F}^{ * } \) successively in this order, while at the same time constructing \( {G}^{ * } \) . 28. Solve the previous exercise first. 29. Use the bijections that come with the two duals to define the desired isomorphism and to prove that it is combinatorial. 30. Apply Menger's theorem and Proposition 4.6.1. For (iii), consider a 4-connected graph with six vertices. 31. Apply induction on \( n \), starting with part (i) of the previous exercise. 32. Theorem 1.9.5. 33. This can be proved directly, i.e. without planarity. ## Hints for Chapter 5 1. Duality. 2.- Whenever more than three countries have some point in common, apply a small local change to the map where this happens. 3. Where does the five colour proof use the fact that \( v \) has no more neighbours than there are colours? 4. How can the colourings of different blocks interfere with each other? 5. \( {}^{ - } \) Use a colouring of \( G \) to derive a suitable ordering. 6. Consider how the removal of certain edges may lead the greedy algorithm to use more colours. 7. Describe more precisely how to implement this alternative algorithm. Then, where is the difference to the traditional greedy algorithm? 8. Compare the number of edges in a subgraph \( H \) as in Proposition 5.2.2 with the number \( m \) of edges in \( G \) . 9. Go via minimum degrees. 10. \( {}^{ - } \) Remove vertices successively until the graph becomes critically \( k \) - chromatic. What can you say about the degree of any vertex that remains? 11. Proposition 1.6.1. 12. \( {}^{ + } \) Modify colourings of the two sides of a hypothetical cut of fewer than \( k - 1 \) edges so that they combine to a \( \left( {k - 1}\right) \) -colouring of the entire graph (with a contradiction). 13. Proposition 1.3.1. 14. \( {}^{ - } \) For which graphs with large maximum degree does Proposition 5.2.2 give a particularly small upper bound? 15. \( {}^{ + } \) (i) How will \( {v}_{1} \) and \( {v}_{2} \) be coloured, and how \( {v}_{n} \) ? (ii) Consider the subgraph induced by the neighbours of \( {v}_{n} \) . 16. \( {}^{ + } \) For the implication (ii) \( \rightarrow \) (i), consider a maximal spanning directed subgraph \( D \) of the given orientation of \( G \) that contains no directed cycle. Use the fact that all directed paths in \( D \) are short to \( k \) -colour its underlying undirected graph, and show that this colouring is even a \( k \) -colouring of \( G \) . 17. In the induction step, compare the values of \( {P}_{G}\left( k\right) ,{P}_{G - e}\left( k\right) \) and \( {P}_{G/e}\left( k\right) \) . 18. \( {}^{ + } \) Multiplicities of zeros. 19. Imitate the proof of Theorem 5.2.6. 20. \( {K}_{n, n} \) . 21. How are edge colourings related to matchings? 22. Construct a bipartite \( \Delta \left( G\right) \) -regular graph that contains \( G \) as subgraph. It may be necessary to add some vertices. 23. \( {}^{ + } \) Induction on \( k \) . In the induction step \( k \rightarrow k + 1 \), consider using several copies of the graph you found for \( k \) . 24. \( {}^{ - } \) Vertex degrees. 25. \( {K}_{n, n} \) . To choose \( n \) so that \( {K}_{n, n} \) is not even \( k \) -choosable, consider lists of \( k \) -subsets of a \( {k}^{2} \) -set. 26. \( {}^{ - } \) Vizing’s theorem. 27. All you need are the definitions, Proposition 5.2.2, and a standard argument from Chapter 1.2. 28. \( {}^{ + } \) Try induction on \( r \) . In the induction step, you would like to to delete one pair of vertices and only one colour from the other vertices' lists. What can you say about the lists if this is impossible? This information alone will enable you to find a colouring directly, without even looking at the graph again. 29. Show that \( {\chi }^{\prime \prime }\left( G\right) \leq {\operatorname{ch}}^{\prime }\left( G\right) + 2 \), and use this to deduce \( {\chi }^{\prime \prime }\left( G\right) \leq \) \( \Delta \left( G\right) + 3 \) from the list colouring conjecture. 30. \( {}^{ - } \) For the first question, try to construct an oriented graph without a kernel edge by edge. For the second and third question, recall the motivational remarks in the text concerning the notion of a kernel. 31. \( {}^{ + } \) Call a set \( S \) of vertices in a directed graph \( D \) a core if \( D \) contains a directed \( v - S \) path for every vertex \( v \in D - S \) . If, in addition, \( D \) contains no directed path between any two vertices of \( S \), call \( S \) a strong core. Show first that every core contains a strong core. Next, define inductively a partition of \( V\left( D\right) \) into ’levels’ \( {L}_{0},\ldots ,{L}_{n} \) such that, for even \( i,{L}_{i} \) is a suitable strong core in \( {D}_{i} \mathrel{\text{:=}} D - \left( {{L}_{0} \cup \ldots \cup {L}_{i - 1}}\right) \), while for odd \( i,{L}_{i} \) consists of the vertices of \( {D}_{i} \) that send an edge to \( {L}_{i - 1} \) . Show that, if \( D \) has no directed odd cycle, the even levels together form a kernel of \( D \) . 32. Construct the orientation needed for Lemma 5.4.3 in steps: if, in the current orientation, there are still vertices \( v \) with \( {d}^{ + }\left( v\right) \geq 3 \), reverse the directions of an edge at \( v \) and take care of the knock-on effect of this change. If you need to bound the average degree of a bipartite planar graph, remember Euler's formula. 33. \( {}^{ - } \) Start with a non-perfect graph. 34. \( {}^{ - } \) Do odd cycles or their complements satisfy \( \left( *\right) \) ? 35. Apply the property of \( {\mathcal{H}}_{1} \) to the graphs in \( {\mathcal{H}}_{2} \), and vice versa. 36. König's theorem asserts the existence of a set of vertices meeting every edge. Rephrase perfection as asserting the existence of a set of vertices meeting all colour classes. 37. Look at the complement. 38. Define the colour classes of a given induced subgraph \( H \subseteq G \) inductively, starting with the class of all minimal elements. 39. (i) Can the vertices on an induced cycle contain each other as intervals? (ii) Use the natural ordering of the reals. 40. Compare \( \omega \left( H\right) \) with \( \Delta \left( G\right) \) (where \( H = L\left( G\right) \) ). 41. \( {}^{ + } \) Which graphs are such that their line graphs contain no induced cycles of odd length \( \geq 5 \) ? To prove that the edges of such a graph \( G \) can be coloured with \( \omega \left( {L\left( G\right) }\right) \) colours, imitate the proof of Vizing’s theorem. 42. Use \( A \) as a colour class. 43. \( {}^{ + } \) (i) Induction. (ii) Assume that \( G \) contains no induced \( {P}^{3} \) . Suppose some \( H \) has a maximal complete subgraph \( K \) and a maximal set \( A \) of independent vertices disjoint from \( K \) . For each vertex \( v \in K \), consider the set of neighbours of \( v \) in \( A \) . How do these sets intersect? Is there a smallest one? 44. \( {}^{ + } \) Start with a candidate for the set \( \mathcal{O} \), i.e. a set of maximal complete subgraphs covering the vertex set of \( G \) . If all the elements of \( \mathcal{O} \) happen to have order \( \omega \left( G\right) \), how does the existence of \( \mathcal{A} \) follow from the perfection of \( G \) ? If not, can you expand \( G \) (maintaining perfection) so that they do and adapt the \( \mathcal{A} \) for the expanded graph to \( G \) ? 45. \( {}^{ + } \) Reduce the general case to the case when all but one of the \( {G}_{x} \) are trivial; then imitate the proof of Lemma 5.5.5. ## Hints for Chapter 6 1. \( {}^{ - } \) Move the vertices, one by one, from \( \bar{S} \) to \( S \) . How does the value of \( f\left( {S,\bar{S}}\right) \) change each time? 2. (i) Trick the algorithm into repeatedly using the middle edge in alternating directions. (ii) At any given time during the algorithm, consider for each vertex \( v \) the shortest \( s - v \) walk that qualifies as an initial segment of an augmenting path. Show for each \( v \) that the length of this \( s - v \) walk never decreases during the algorithm. Now consider an edge which is used twice for an augmenting path, in the same direction. Show that the second of these paths must have been longer than the first. Now derive the desired bound. 3. \( {}^{ + } \) For the edge version, define the capacity function so that a flow of maximum value gives rise to sufficie

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\( {}^{ + } \) Reduce the general case to the case when all but one of the \( {G}_{x} \) are trivial; then imitate the proof of Lemma 5.5.5. ## Hints for Chapter 6 1. \( {}^{ - } \) Move the vertices, one by one, from \( \bar{S} \) to \( S \) . How does the value of \( f\left( {S,\bar{S}}\right) \) change each time? 2. (i) Trick the algorithm into repeatedly using the middle edge in alternating directions. (ii) At any given time during the algorithm, consider for each vertex \( v \) the shortest \( s - v \) walk that qualifies as an initial segment of an augmenting path. Show for each \( v \) that the length of this \( s - v \) walk never decreases during the algorithm. Now consider an edge which is used twice for an augmenting path, in the same direction. Show that the second of these paths must have been longer than the first. Now derive the desired bound. 3. \( {}^{ + } \) For the edge version, define the capacity function so that a flow of maximum value gives rise to sufficiently many edge-disjoint paths. For the vertex version, split every vertex \( x \) into two adjacent vertices \( {x}^{ - },{x}^{ + } \) . Define the edges of the new graph and their capacities in such a way that positive flow through an edge \( {x}^{ - }{x}^{ + } \) corresponds to the use of \( x \) by a path in \( G \) . 4. \( {}^{ - }H \) -flows are nowhere zero, by definition. 5. \( {}^{ - } \) Use the definition and Proposition 6.1.1. 6. \( {}^{ - } \) Do subgraphs also count as minors? 7. \( {}^{ - } \) Try \( k = 2,3,\ldots \) in turn. In searching for a \( k \) -flow, tentatively fix the flow value through an edge and investigate which consequences this has for the adjacent edges. 8. To establish uniqueness, consider cuts of a special type. 9. Express \( G \) as the union of cycles. 10. Combine \( {\mathbb{Z}}_{2} \) -flows on suitable subgraphs to a flow on \( G \) . 11. Begin by sending a small amount of flow through every edge outside \( T \) . 12. View \( G \) as the union of suitably chosen cycles. 13. Corollary 6.3.2 and Proposition 6.4.1. 14. \( {}^{ - } \) Duality. 15. Take as \( H \) your favourite graph of large flow number. Can you decrease its flow number by adding edges? 16. Euler. 17. Instead of proving (F2) for \( g \), show more generally that \( g\left( {X,\bar{X}}\right) = 0 \) for every cut \( {E}^{ * }\left( {X,\bar{X}}\right) \) of \( {G}^{ * } \) . 18. \( {}^{ + } \) Theorem 6.5.3. 19. Theorem 6.5.3. 20. (i) Theorem 6.5.3. (ii) Yes it can. Show, by considering a smallest counterexample, that if every 3-connected cubic planar multigraph is 3-edge-colourable (and hence has a 4-flow), then so is every bridgeless cubic planar multigraph. 21. \( {}^{ + } \) For the ’only if’ implication apply Proposition 6.1.1. Conversely, consider a circulation \( f \) on \( G \), with values in \( \{ 0, \pm 1,\ldots , \pm \left( {k - 1}\right) \} \), that respects the given orientation (i.e. is positive or zero on the edge directions assigned by \( D \) ) and is zero on as few edges as possible. Then show that \( f \) is nowhere zero, as follows. If \( f \) is zero on \( e = {st} \in E \) and \( D \) directs \( e \) from \( t \) to \( s \), define a network \( N = \left( {G, s, t, c}\right) \) such that any flow in \( N \) of positive total value contradicts the choice of \( f \), but any cut in \( N \) of zero capacity contradicts the property assumed for \( D \) . 22. \( {}^{ - } \) Convert the given multigraph into a graph with the same flow properties. ## Hints for Chapter 7 1. \( {}^{ - } \) Straightforward from the definition. 2. \( {}^{ - } \) When constructing the graphs, start by fixing the colour classes. 3. It is not difficult to determine an upper bound for \( \operatorname{ex}\left( {n,{K}_{1, r}}\right) \) . What remains to be proved is that this bound can be achieved for all \( r \) and \( n \) . 4. Proposition 1.7.2 (ii). 5. \( {}^{ + } \) What is the maximum number of edges in a graph of the structure given by Theorem 2.2.3 if it has no matching of size \( k \) ? What is the optimal distribution of vertices between \( S \) and the components of \( G - S \) ? Is there always a graph whose number of edges attains the corresponding upper bound? 6. Consider a vertex \( x \in G \) of maximum degree, and count the edges in \( G - x \) . 7. Choose \( k \) and \( i \) so that \( n = \left( {r - 1}\right) k + i \) with \( 0 \leq i < r - 1 \) . Treat the case of \( i = 0 \) first, and then show for the general case that \( {t}_{r - 1}\left( n\right) = \) \( \frac{1}{2}\frac{r - 2}{r - 1}\left( {{n}^{2} - {i}^{2}}\right) + \left( \begin{array}{l} i \\ 2 \end{array}\right) . \) 8. The bounds given in the hint are the sizes of two particularly simple Turán graphs-which ones? 9. Choose among the \( m \) vertices a set of \( s \) vertices that are still incident with as many edges as possible. 10. For the first inequality, double the vertex set of an extremal graph for \( {K}_{s, t} \) to obtain a bipartite graph with twice as many edges but still not containing a \( {K}_{s, t} \) . 11. \( {}^{ + } \) For the displayed inequality, count the pairs \( \left( {x, Y}\right) \) such that \( x \in A \) and \( Y \subseteq B \), with \( \left| Y\right| = r \) and \( x \) adjacent to all of \( Y \) . For the bound on \( \operatorname{ex}\left( {n,{K}_{r, r}}\right) \), use the estimate \( {\left( s/t\right) }^{t} \leq \left( \begin{array}{l} s \\ t \end{array}\right) \leq {s}^{t} \) and the fact that the function \( z \mapsto {z}^{r} \) is convex. 12. Assume that the upper density is larger than \( 1 - \frac{1}{r - 1} \) . What does this mean precisely, and what does the Erdős-Stone theorem then imply? 13. Proposition 1.2.2 and Corollary 1.5.4. 14. Complete graphs. 15. \( {}^{ - } \) A vertex of high degree is nearly a star. 16. Do more than \( \frac{1}{2}\left( {k - 1}\right) n \) edges force a subgraph of suitable minimum degree? 17. \( {}^{ + } \) Consider your favourite graphs with high average degree and low chromatic number. Which trees do they contain induced? Is there some reason to expect that exactly these trees may always be found induced in graphs of large average degree and small chromatic number? 18. All the implications sought are either very easy to prove or follow from material stated in the text (not necessarily in this chapter). 19. \( {}^{ + } \) Contract a set of the form \( \left\{ {v \mid d\left( {{v}_{0}, v}\right) \leq i}\right\} \) . 20. Induction on \( r \) . 21. \( {}^{ - } \) Does a large chromatic number force up the average degree? If in doubt, consult Chapter 5. 22. \( {}^{ + } \) Let \( {G}^{\prime } \preccurlyeq G \) be a minimal minor with \( \varepsilon \left( {G}^{\prime }\right) \geq k \) . Show that, for every vertex \( v \in {G}^{\prime } \), the subgraph \( H \) of \( {G}^{\prime } \) induced by the neighbours of \( v \) has minimum degree at least \( k \) . Can you choose \( v \) so that \( \left| H\right| \leq {2k} \) ? 23. \( {}^{ + } \) First show that we need only consider graphs \( G \) of minimum degree at least 3. Then Corollary 1.3.5 gives us a cycle \( C \) of length at most about \( 2\log n \) . Assuming without loss of generality that \( G \) has exactly \( n + \lceil {2k}\left( {\log k + \log \log k + c}\right) \rceil \geq {3n}/2 \) edges, bound \( \parallel C\parallel \) from above in terms of \( k \), and show that, for a suitable choice of \( c \), deleting only this many edges makes the induction step work. 24. \( {}^{ - } \) Imitate the proof of Theorem 7.2.1, replacing \( {r}^{2} \) by \( \left( \begin{array}{l} r \\ 2 \end{array}\right) \) . 25. \( {}^{ + } \) How can we best make a \( T{K}^{2r} \) fit into a \( {K}_{s, s} \) when we want to keep \( s \) small? 26. Which of the graphs constructed as in the hint have the largest average degree? 27. \( {}^{ - } \) What does planarity have to do with minors? 28. \( {}^{ - } \) Consider a suitable supergraph. 29. \( {}^{ - } \) Apply a theorem from this chapter. 30. Induction on the number of construction steps. 31. Induction on \( \left| G\right| \) . 32. Note the previous exercise. 33. Start with a suitable subgraph of large minimum degree. Which result or technique from Section 7.2 can be used to boost its minimum degree further to make suitable input for Theorem 7.2.2? 34. \( {}^{ + } \) Show by induction on \( \left| G\right| \) that any 3-colouring of an induced cycle in \( G \nsucceq {K}^{4} \) extends to all of \( G \) . 35. \( {}^{ + } \) Reduce the statement to critical \( k \) -chromatic graphs and apply Vizing’s theorem. 36. Which of the graphs constructed as in Theorem 7.3.4 have the largest average degree? 37. \( {}^{ - } \) Why would it be impractical to include, say,1-element sets \( X, Y \) in the comparison? 38. \( {}^{ - } \) Apply the definition of an \( \epsilon \) -regular pair. 39. For the meaning of the word ’about’, assume that \( \left| V\right| \) is large compared with \( k \) . For the second task, do not refer to the details of the proof of Theorem 7.1.2, but to the informal explanations follows it. 40. For (i) just make \( M \) large enough. For (ii) use the analogue of (i) for the graphs considered, putting \( k \mathrel{\text{:=}} m \) when the graph is large. ## Hints for Chapter 8 1. \( {}^{ - } \) Count the vertices,’moving out’ from a fixed vertex. 2.- Make \( \sigma \) beat \( {\sigma }^{i} \) from \( {s}_{i} \) onwards. 3. Let \( \mathcal{A} \) be a set of subsets of a countable set \( A \) such that \( \left| {{A}^{\prime } \cap {A}^{\prime \prime }}\right| \leq k \) for all distinct \( {A}^{\prime },{A}^{\prime \prime } \in \mathcal{A} \) and some fixed \( k \in \mathbb{N} \) . Consider a fixed \( k \) -set \( S \) . How many sets in \( \mathcal{A} \) can contain \( S \) ? 4. \( {}^{ - } \) Consider a ray \( {v}_{0}{v}_{1}\ldots \) Can it be decreasing, ie such that \( {v}_{0} > {v}_{1} > \) \( \ldots \) ? If not, ca

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compared with \( k \) . For the second task, do not refer to the details of the proof of Theorem 7.1.2, but to the informal explanations follows it. 40. For (i) just make \( M \) large enough. For (ii) use the analogue of (i) for the graphs considered, putting \( k \mathrel{\text{:=}} m \) when the graph is large. ## Hints for Chapter 8 1. \( {}^{ - } \) Count the vertices,’moving out’ from a fixed vertex. 2.- Make \( \sigma \) beat \( {\sigma }^{i} \) from \( {s}_{i} \) onwards. 3. Let \( \mathcal{A} \) be a set of subsets of a countable set \( A \) such that \( \left| {{A}^{\prime } \cap {A}^{\prime \prime }}\right| \leq k \) for all distinct \( {A}^{\prime },{A}^{\prime \prime } \in \mathcal{A} \) and some fixed \( k \in \mathbb{N} \) . Consider a fixed \( k \) -set \( S \) . How many sets in \( \mathcal{A} \) can contain \( S \) ? 4. \( {}^{ - } \) Consider a ray \( {v}_{0}{v}_{1}\ldots \) Can it be decreasing, ie such that \( {v}_{0} > {v}_{1} > \) \( \ldots \) ? If not, can it go down again once it has gone up, ie, can it contain vertices \( {x}_{i - 1} < {x}_{i} > {x}_{i + 1} \) ? 5. \( {}^{ - } \) Construct the paths inductively. Alternatively, use Zorn’s lemma to find a maximal set of disjoint \( A - B \) paths. Can it be finite? 6. \( {}^{ - } \) If you cannot make this approach work, describe how it fails. 7. \( {}^{ - } \) Construct such a graph inductively. Can you do it in one infinite sequence of steps? 8. Construct the graph inductively, starting from a vertex or a cycle. To ensure that the final graph has high connectivity, join each new vertex by many edges to the infinite set of vertices yet to be defined. 9. \( {}^{ - } \) Use the previous exercise. 10. Starting from the definition of the topology on \( X \), describe what it means for a sequence of points in \( X \) to converge. What must a sequence look like whose convergent subsequences all determine proper colourings of \( G \) ? Can you deduce from the assumptions that such a sequence exists? It may help to look at the infinity lemma for ideas. 11. \( {}^{ + } \) Apply induction on \( k \) . 12. \( {}^{ - } \) This is a standard compactness proof: use the infinity lemma for countable graphs, and Tychonov's theorem for arbitrary graphs. 13. Apply the infinity lemma. Find a statement about a vertex partition of \( {G}_{n} = G\left\lbrack {{v}_{1},\ldots ,{v}_{n}}\right\rbrack \) that implies the corresponding statement for the induced partition of \( {G}_{n - 1} \), and whose truth for the partitions of the \( {G}_{n} \) induced by a given partition of \( G \) implies that this partition of \( G \) is as desired. 14. Apply the infinity lemma to a suitably weakened statement about finite subgraphs. 15. For the positive result use the infinity lemma, considering the finite subgraphs spanned by a given finite subset of \( A \) and all its neighbours in \( B \) . For the counterexample, note that if \( S \subsetneqq {S}^{\prime } \subseteq A \) with \( \left| S\right| = \left| {N\left( S\right) }\right| \) in the finite case, the marriage condition ensures that \( N\left( S\right) \subsetneqq N\left( {S}^{\prime }\right) \) : increasing \( S \) makes more neighbours available. Use the fact that this can fail when \( S \) is infinite. 16. \( {}^{ + } \) Note that, in order to apply the infinity lemma, it is enough to find in every finite induced subgraph \( {G}_{n} \) of \( G \) a set of independent edges covering those vertices that have no neighbour in \( G - {G}_{n} \) . To find such a set of edges, apply the finite 1-factor theorem to the graph \( {H}_{n} \) obtained from \( {G}_{n} \) by adding a large complete graph \( K \) joined completely to all those vertices of \( {G}_{n} \) that have a neighbour in \( G - {G}_{n} \) . If you get stuck, change the parity of \( \left| K\right| \) . 17. \( {}^{ + } \) Use the material from Chapter 4.3 to make drawings susceptible to an application of the infinity lemma. To construct the final drawing from a ray in the infinity lemma graph, make sure that the partial drawings constructed inductively are really definite drawings in the plane, not merely abstract equivalence types of drawings. 18. \( {}^{ - } \) Adapt the hint for Exercise 5 to prove the appropriate fan version of Menger's theorem. 19. Construct the \( T{K}^{{\aleph }_{0}} \) inductively. 20. Start with the binary tree \( {T}_{2} \), and make its ends thick while keeping the graph countable. 21. You can prove the forward implication either 'from above' by recursively pruning away parts of the tree that are certain not to lie in a subdivided \( {T}_{2} \), or ’from below’ by constructing a subdivided \( {T}_{2} \) inductively inside the given tree. 22. For (i), note that a ray has countably many subrays. For the forward implication in (iii), prune the given tree recursively by chopping off locally finite subtrees and bounding these; then combine all the bounding functions obtained into one. It will help in the proof if you make this final function increasing. 23. \( {}^{ + } \) Does \( {T}_{{\aleph }_{1}} \) have such a labelling? If \( T \nsupseteq T{T}_{{\aleph }_{1}} \), construct a labelling of \( T \) inductively. Supposing a labelling exists: where in \( T \) will the vertices labelled zero lie? Where the vertices labelled 1 ? 24. Suppose a locally finite connected graph \( G \) has three distinct ends. Let \( S \) be a finite set of vertices separating these pairwise. Take an automorphism that maps \( S \) ’far away’ into a component of \( G - S \) . Can you show that the image of \( S \) separates this component in such a way that \( G \) must have more than three ends? 25. \( {}^{ + } \) Pick a vertex \( v \) . Is its orbit \( U = \{ v,\sigma \left( v\right) ,\sigma \left( {\sigma \left( v\right) }\right) ,\ldots \} \) finite or infinite? To determine the position of \( U \) within \( G \), let \( P \) be a path from \( v \) to \( \sigma \left( v\right) \) and consider the infinite union \( P \cup \sigma \left( P\right) \cup \sigma \left( {\sigma \left( P\right) }\right) \cup \ldots \) . Does this, somehow, define an end? And what about the sequence \( v,{\sigma }^{-1}\left( v\right) ,{\sigma }^{-2}\left( v\right) ,\ldots \) ? 26. \( {}^{ - } \) Lemma 8.2.2. 27. Lemma 8.2.3. 28. Prove the implication (i) \( \rightarrow \) (iv) first. 29. Show that deleting a finite set of vertices never leaves infinitely many components. 30. To construct the normal spanning tree in (i), imitate the proof for countable \( G \) . Well-order each of the dispersed sets, concatenate these well-orderings into one well-ordering of \( V\left( G\right) \), and construct the tree recursively. 31. \( {}^{ - } \) Normal spanning trees. 32. \( {}^{ + } \) For simplicity, replace the graph with a spanning tree in it, \( T \) say. Which vertices have to appear earlier in the enumeration than others? 33. Imitate the proof of Theorem 8.2.5, choosing all the rays used from the given end. Do the rays constructed also belong to that end? If not, how can this be achieved? 34. \( {}^{ + } \) Imitate the proof of Theorem 8.2.5. Work with rays rather than double rays whenever possible. 35. The task is to find in any graph \( G \) that contains arbitrarily many disjoint \( {MH} \) a locally finite subgraph with the same property. In a first step, find a countable such subgraph \( {G}^{\prime } \), and enumerate its vertices. Then use the enumeration to find a locally finite such subgraph \( {G}^{\prime \prime } \subseteq {G}^{\prime } \) by ensuring that each vertex of \( {G}^{\prime } \) is used by only finitely many \( {MH} \) . 36. To construct a graph that contains arbitrarily but not infinitely many copies of the modified comb \( T \), start with infinitely many disjoint copies of \( T \) . Group these into disjoint sets \( {S}_{1},{S}_{2},\ldots \) so that \( {S}_{n} \) is a disjoint union of \( n \) copies of \( T \) . Then identify vertices from different sets \( {S}_{n} \), so as to spoil infinite ’diagonal’ sets of disjoint copies of \( T \) . 37. Fundamental cycles. 38. Unlike in the proof of Theorem 8.2.6, you can use suitable tails of all the rays in the (large but finite) set \( {\mathcal{R}}_{0} \) as rays \( {Q}_{n} \) . The part of the proof that start with assumption \( \left( *\right) \) can thus be replaced by a much simpler algorithm that finds \( {Q}_{n} \) and an infinite set of disjoint \( {Q}_{n} - {Q}_{p\left( n\right) } \) paths. To determine how many rays are needed, start with a suitable finite analogue to the infinity lemma: any large enough rooted tree either has a vertex with at least \( k \) successors or contains a path of length \( k \) . 39. Suppose there is a universal graph \( G \) . Construct a locally finite connected graph \( H \) whose vertex degrees ’grow too fast’ for any embedding of \( H \) in \( G \) . 40. Modify \( {K}^{{\aleph }_{0}} \) or the Rado graph. Or try a direct construction. 41. \( {}^{ - } \) Property \( \left( *\right) \) . 42. Back-and-forth. 43. Back-and-forth. 44. Find the partition inductively, deleting the edge set of one graph at a time and showing that what remains is still isomorphic to \( R \) . How can you ensure that, once all the required edge sets have been deleted, there is no edge left? 45. \( {}^{ - } \) This is a theorem of Cantor. To prove it, use density like property \( \left( *\right) \) . 46. \( R \) . 47. For the vertex \( v \) in property \( \left( *\right) \), try putting \( v \mathrel{\text{:=}} U \) first. How can this fail? And how can you amend it if it fails? You may wish to use the Axiom of Foundation, by which there is no sequence \( {x}_{1} \in \ldots \in {x}_{n} \) of sets with \( n \geq 2 \) and \( {x}_{1} = {x}_{n} \) . 48. Look at Exercise 49 and its hint. For locally finite \( G \) the sets \( {S}_{i}^{\prime } \) are very easy to find, and no normal spanning tree is needed. 49. \( {}^{ + } \) Use a normal spanning tree

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- } \) Property \( \left( *\right) \) . 42. Back-and-forth. 43. Back-and-forth. 44. Find the partition inductively, deleting the edge set of one graph at a time and showing that what remains is still isomorphic to \( R \) . How can you ensure that, once all the required edge sets have been deleted, there is no edge left? 45. \( {}^{ - } \) This is a theorem of Cantor. To prove it, use density like property \( \left( *\right) \) . 46. \( R \) . 47. For the vertex \( v \) in property \( \left( *\right) \), try putting \( v \mathrel{\text{:=}} U \) first. How can this fail? And how can you amend it if it fails? You may wish to use the Axiom of Foundation, by which there is no sequence \( {x}_{1} \in \ldots \in {x}_{n} \) of sets with \( n \geq 2 \) and \( {x}_{1} = {x}_{n} \) . 48. Look at Exercise 49 and its hint. For locally finite \( G \) the sets \( {S}_{i}^{\prime } \) are very easy to find, and no normal spanning tree is needed. 49. \( {}^{ + } \) Use a normal spanning tree to find provisional sets \( {S}_{1}^{\prime },{S}_{2}^{\prime },\ldots \) of arbitrary finite cardinality that have the separation properties required of the \( {S}_{i} \) . Then use these to find the \( {S}_{i} \) . 50. Pick \( a \in A \), and construct a sequence of waves \( {\mathcal{W}}_{1},{\mathcal{W}}_{2},\ldots \) that each contain the trivial path \( \{ a\} \) . Define the edges at \( a \) so that \( a \) is in the boundary of every \( {\mathcal{W}}_{n} \), but not in the boundary of the limit wave. 51. \( {}^{ + } \) The general problem reduces to Lemma 8.4.3, just as in the countable case. Prove the lemma for forests. 52. \( {}^{ + } \) Starting with \( \mathcal{P} \), recursively define path systems \( {\mathcal{P}}_{\alpha } \) that link \( A \) to more and more of \( B \) . In the recursion step, pick an uncovered vertex \( b \in B \) and follow the path \( Q \in \mathcal{Q} \) containing it back until it hits \( {\mathcal{P}}_{\alpha } \) , say in \( P = a\ldots {b}^{\prime } \) . You could then re-route \( P \) to follow \( Q \) to \( b \) from there, but this would leave \( {b}^{\prime } \) uncovered. Still, could it be that these changes produce an increase of the covered part of \( B \) at limit steps? To prove that it does, can you define an 'index' parameter that grows (or decreases) with every step but cannot do so indefinitely? Alternatively, prove and apply a suitable infinite version of the stable marriage theorem (2.1.4). 53. (i) is just compactness. A neat 1-line proof uses Theorem 8.1.3. For (ii), construct a poset from arbitrarily large finite antichains. For (iii), define a bipartite graph as follows. For every point \( x \in P \) take two vertices \( {x}^{\prime } \) and \( {x}^{\prime \prime } \) . Then add all edges \( {x}^{\prime }{y}^{\prime \prime } \) such that \( x < y \) . Now consider a matching \( M \) and a vertex cover \( U \) in this graph as provided by Theorem 8.4.8. How does \( M \) define a partition of \( P \) into chains? For how many points \( x \) of such a chain can \( {x}^{\prime } \) or \( {x}^{\prime \prime } \) lie in \( U \) ? 54. Do the assumptions imply that there exists a 1-factor? If so, can you use it? 55. To ensure that every partial matching can be augmented, give your graph lots of edges. How can you nevertheless prevent a 1-factor? 56. Try to prove, e.g. by compactness, that an infinite factor-critical graph must have a 1-factor. If your proof fails, does it lead you to a construction? 57. \( {}^{ - } \) Consider first the case that the complete subgraphs of \( H \) have finitely bounded order. You may use a result from Section 8.1. 58. \( {}^{ + } \) For the perfection of \( G \) in (ii), show that every subset of \( {T}_{2} \) with arbitrarily large finite antichains also has an infinite antichain. 59. For the backwards implication, note that no finite set of vertices separates \( R \) from \( X \) . Use this to construct the \( R - X \) paths inductively, or apply a trivial version of Menger's theorem. 60. A sequentially compact space (one where every infinite sequence of points has a convergent subsequence) is compact if it has a countable basis. If the infinity lemma does not seem to help, look at Lemma 8.2.2. 61. \( {}^{ + } \) For the compactness proof, use a normal spanning tree and imitate the proof of Proposition 8.5.1. 62. Your answer may depend on whether \( H \) is known to be locally finite. Remember that a continuous bijection from a compact space to Hausdorff space is a homeomorphism. For (iii), you may use a theorem from the text. 63. \( {}^{ + } \) For the first task, scale the lengths of the edges of the tree down to ensure that the total length of a ray starting at the root becomes finite. Then adjust the lengths of the other edges of \( G \), and extend the metric obtained to the ends of \( G \) . For the second, notice that for any metric inducing the given topology on \( V \cup \Omega \) the sets \( {V}_{n} \) of vertices at distance at least \( 1/n \) from every end are closed, and show that these sets cover \( V \) as \( n \) ranges over the positive integers. 64. \( {}^{ + } \) To define the topology on \( \widehat{X} \), imitate the definition of the usual one-point compactification. 65. You may use that deleting an open interval from the unit circle leaves a connected rest, but that deleting two disjoint open intervals does not. Remember that closed connected subsets of \( \left| G\right| \) are path-connected. 66. Construct two rays that belong to the same end and start at the same vertex but are otherwise disjoint. This can be done by considering a normal ray and using the fact that none of its vertices is a cutvertex. 67. Recall that, in \( {S}^{1} \), every point has a neighbourhood basis consisting of \( \arcsin {\mathbb{R}}^{2} \) . Can you show that every arc in \( C \) that links two ends must meet an edge? If not, can you show that it meets a vertex? If not, remember the proof of Lemma 8.5.5. 68. Exercise 26. 69. Enumerate the double rays \( D \) and \( {D}_{\ell } \) in one infinite sequence, and inductively define partial homeomorphisms between these \( {D}_{\ell } \) and suitable segments of \( {S}^{1} \) . When this is done, extend the partial homeomorphism on the union of all the double rays to the ends of \( G \) so as to make the final map continuous. 70. The main assertion to be proved is that every subspace \( C \) satisfying the conditions is a circle. Let \( A \subseteq C \) be an arc linking two vertices \( {x}_{0} \) and \( {y}_{0} \) . If \( v \) is any vertex in \( C \smallsetminus A \), the arc-connectedness of \( C \) yields a \( v - A \) arc in \( C \), which has a first point on \( A \) . By the degree condition assumed, this must be \( {x}_{0} \) or \( {y}_{0} \) . Starting from an enumeration \( {v}_{0},{v}_{1},\ldots \) of the vertices in \( C \), construct a 2-way infinite sequence \( \ldots {x}_{-2},{x}_{-1},{x}_{0},{y}_{0},{y}_{1},{y}_{2}\ldots \) of vertices such that \( C \) contains arcs \( {A}_{i} \) linking \( {x}_{-i - 1} \) to \( {x}_{-i} \) and \( {B}_{i} \) linking \( {y}_{i} \) to \( {y}_{i + 1} \) for all \( i \in \mathbb{N} \), so that the union \( U \) of \( A \) and all these arcs is a homeomorphic copy of \( \left( {0,1}\right) \) in \( C \) . Use the connectedness of its ’tails’ to show that these converge to unique ends in \( C \) . Deduce from the degree assumptions that these two ends coincide, and that \( \bar{U} = C \) is a circle. 71. Use Lemma 8.5.4. You may also use that every circle contains an edge. 72. \( {}^{ - } \) Show that if a topological spanning tree is homeomorphic to a space \( \left| T\right| \) with \( T \) a tree, but does not itself have this form, it contains an end which this homeomorphism maps to a point in \( T \) (i.e., not to an end). Can you find a topological spanning tree for which this is impossible? 73. Start with a maximal set of disjoint rays. 74. \( {}^{ + } \) Given a point \( \omega \in \bar{A} \smallsetminus A \), pick a sequence \( {v}_{1},{v}_{2},\ldots \) of vertices in \( A \) that converges to \( \omega \), and arcs \( {A}_{n} \subseteq A \) from \( {v}_{n} \) to \( {v}_{n + 1} \) . Then use the infinity lemma to concatenate suitable portions of the \( {A}_{n} \) to form a continuous function \( \alpha : \left\lbrack {0,1}\right\rbrack \rightarrow \left| G\right| \) that maps \( \lbrack 0,1) \) to \( A \) and 1 to \( \omega \) . You may use the fact that the image of such a function \( \alpha \) contains an arc from \( \alpha \left( 0\right) \in A \) to \( \alpha \left( 1\right) = \omega \) . 75. Recall that non-separating induced cycles of a plane graph are face boundaries. 76. \( {}^{ - } \) How can \( \bar{T} \) fail to be a topological spanning tree? 77. Find the circuits greedily, making sure all edges are captured. 78. Check thinness. For an alternative proof, use Theorem 8.5.8 (i) instead of (ii). 79. \( {}^{ + } \) For the ’only if’ part, use a theorem from the text. The task in the 'if' part is to combine the edge-disjoint circles from Theorem 8.5.8 (ii) into a single continuous image of \( {S}^{1} \) . Start with one of those circles, and incorporate the others step by step. Check that the 'limit map' \( \sigma : {S}^{1} \rightarrow \left| G\right| \) is continuous (and defined) on all of \( {S}^{1} \) . 80. \( {}^{ + } \) The conditions are easily seen to be necessary. To prove sufficiency, construct an Euler tour inductively, incorporating at once any finite components arising in the remaining graph. To ensure that all edges get included, enumerate them, and always target the next edge for inclusion. There are two cases. If \( G \) has an odd cut, cover those edges first, join up their endvertices in pairs as far as possible, and proceed separately in the two infinite components of the rest. If \( G \) has no odd cut, cover its edges inductively by a sequence of finite closed walks, so that each of

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’only if’ part, use a theorem from the text. The task in the 'if' part is to combine the edge-disjoint circles from Theorem 8.5.8 (ii) into a single continuous image of \( {S}^{1} \) . Start with one of those circles, and incorporate the others step by step. Check that the 'limit map' \( \sigma : {S}^{1} \rightarrow \left| G\right| \) is continuous (and defined) on all of \( {S}^{1} \) . 80. \( {}^{ + } \) The conditions are easily seen to be necessary. To prove sufficiency, construct an Euler tour inductively, incorporating at once any finite components arising in the remaining graph. To ensure that all edges get included, enumerate them, and always target the next edge for inclusion. There are two cases. If \( G \) has an odd cut, cover those edges first, join up their endvertices in pairs as far as possible, and proceed separately in the two infinite components of the rest. If \( G \) has no odd cut, cover its edges inductively by a sequence of finite closed walks, so that each of these meets the next in a vertex. Then find an Euler tour in the union of these cycles. ## Hints for Chapter 9 1. \( {}^{ - } \) Can you colour the edges of \( {K}^{5} \) red and green without creating a red or a green triangle? Can you do the same for a \( {K}^{6} \) ? 2. \( {}^{ - } \) Induction on \( c \) . In the induction step, unite two of the colour classes. 3. If the chromatic number of a graph is small, does this imply the existence of a large induced \( \overline{{K}^{r}} \) ? If so, how large? 4. \( {}^{ + } \) Choose a well-ordering of \( \mathbb{R} \), and compare it with the natural ordering. Use the fact that countable unions of countable sets are countable. 5. \( {}^{ + } \) Suppose there are many chords \( {xy} \), with \( x{ < }_{T}y \) say, whose paths \( {xTy} \) meet pairwise in at least one edge. Find either a large set of such vertices \( x \) whose partner vertices \( y \) coincide, or a vertex in \( T \) with many incomparable vertices \( y \) above it, or a long ascending path in \( T \) whose maximal vertices \( {t}_{y} \) on \( {xTy} \) are distinct for many \( y \) . Then find a long sequence \( {x}_{1} \leq \ldots \leq {x}_{n} \) of vertices \( x \) corresponding to these \( y \), and show that the union of the paths \( {x}_{i}T{y}_{i} \) together with the chords \( {x}_{i}{y}_{i} \) contains many edge-disjoint cycles. 6. \( {}^{ + } \) The first and second question are easy. To prove the theorem of Erdős and Szekeres, use induction on \( k \) for fixed \( \ell \), and consider in the induction step the last elements of increasing subsequences of length \( k \) . Alternatively, apply Dilworth's Theorem. 7. Use the fact that \( n \geq 4 \) points span a convex polygon if and only if every four of them do. 8. Translate the given \( k \) -partition of \( \{ 1,2,\ldots, n\} \) into a \( k \) -colouring of the edges of \( {K}^{n} \) . 9. (i) is easy. For (ii) use the existence of \( R\left( {2, k,3}\right) \) . 10. Begin by finding infinitely many sets whose pairwise intersections all have the same size. 11. The exercise offers more information than you need. Consult Chapter 7.2 to see what is relevant. 12. Imitate the proof of Proposition 9.2.1. 13. The lower bound is easy. Given a colouring for the upper bound, consider a vertex and the neighbours joined to it by suitably coloured edges. 14. \( {}^{ - } \) Given \( {H}_{1} \) and \( {H}_{2} \), construct a graph \( H \) for which the \( G \) of Theorem 9.3.1 satisfies \( \left( *\right) \) . 15. \( G\left\lbrack {U \rightarrow H}\right\rbrack \) . 16. Show inductively for \( k = 0,\ldots, m \) that \( \omega \left( {G}^{k}\right) = \omega \left( H\right) \) . 17. \( {}^{ - } \) How exactly does Proposition 9.4.1 fail if we delete \( {K}^{r} \) from the statement? 18. As an example, prove that Theorem 9.4.5 (ii) is equivalent to Proposition 9.4.2. The other three equivalences are very similar. ## Hints for Chapter 10 1. Induction. 2. Consider the union of two colour classes. 3. Induction on \( k \) with \( n \) fixed; for the induction step consider \( \bar{G} \) . 4. \( {}^{ - } \) What do \( k \) -connected graphs look like that satisfy \( \chi \left( G\right) \geq \left| G\right| /k \) but not \( \alpha \left( G\right) \leq k \) ? 5. Note that subdividing the edges at a vertex of odd degree is a useful trick to produce non-hamiltonian graphs. To find an example for (ii), apply this trick to a small but highly connected graph. 6. How high can the connectivity of a planar graph be? 7. \( {}^{ - } \) Recall the definition of a hamiltonian sequence. 8. \( {}^{ - } \) On which kind of vertices does the Chvátal condition come to bear? To check the validity of the condition for \( G \), first find such a vertex. 9. Consider a \( k \) -separator in \( {G}^{2} \) . Where do its vertices send their \( G \) -edges? 10. Theorem 10.2.1. 11. How does an arbitrary connected graph differ from the kind of graph whose square contains a Hamilton cycle by Fleischner's theorem? How could this difference obstruct the existence of a Hamilton cycle? 12. \( {}^{ + } \) In the induction step consider a minimal cut. 13. \( {}^{ + } \) How can a Hamilton path \( P \in \mathcal{H} \) be modified into another? In how many ways? What has this got to do with the degree in \( G \) of the last vertex of \( P \) ? ## Hints for Chapter 11 1. \( {}^{ - } \) Consider a fixed choice of \( m \) edges on \( \{ 0,1,\ldots, n\} \) . What is the probability that \( G \in \mathcal{G}\left( {n, p}\right) \) has precisely this edge set? 2. Consider the appropriate indicator random variables, as in the proof of Lemma 11.1.5. 3. Consider the appropriate indicator random variables. 4. Erdős. 5. What would be the measure of the set \( \{ G\} \) for a fixed \( G \) ? 6. Consider the complementary properties. 7. \( {\mathcal{P}}_{2,1} \) . 8. Apply Lemma 11.3.2. 9. Induction on \( \left| H\right| \) with the aid of Exercise 6. 10. Imitate the proof of Lemma 11.2.1. 11. Imitate the proof of Proposition 11.3.1. To bound the probabilities involved, use the inequality \( 1 - x \leq {e}^{-x} \) as in the proof of Lemma 11.2.1. 12. \( {}^{ + } \) (i) Calculate the expected number of isolated vertices, and apply Lemma 11.4.2 as in the proof of Theorem 11.4.3. (ii) Linearity. 13. \( {}^{ + } \) Chapter 7.2, the proof of Erdős’s theorem, and a bit of Chebyshev. 14. For the first problem modify an increasing property slightly, so that it ceases to be increasing but keeps its threshold function. For the second, look for an increasing property whose probability does not really depend on \( p \) . 15. \( {}^{ - } \) Permutations of \( V\left( H\right) \) . 16. \( {}^{ - } \) This is a result from the text in disguise. 17. \( {}^{ - } \) Balance. 18. For \( p/t \rightarrow 0 \) apply Lemmas 11.1.4 and 11.1.5. For \( p/t \rightarrow \infty \) apply Corollary 11.4.4. 19. There are only finitely many trees of order \( k \) . 20. \( {}^{ + } \) Show first that no such threshold function \( t = t\left( n\right) \) can tend to zero as \( n \rightarrow \infty \) . Then use Exercise 11. 21. \( {}^{ + } \) Examine the various steps in the proof of Theorem 11.4.3, identify the two points where it now fails, and repair them. While the first part requires a slightly different tack as a consequence, the second adapts more mechanically. ## Hints for Chapter 12 1. Antisymmetry. 2. For the backward implication, assume first that \( A \) has an infinite an-tichain; this case is easier. The proof for other case is not quite as obvious but similar; note that \( A = \mathbb{Z} \) is not a counterexample. 3. To prove Proposition 12.1.1, consider an infinite sequence in which every strictly decreasing subsequence is finite. How does the last element of a maximal decreasing subsequence compare with the elements that come after it? For Corollary 12.1.2, start by proving that at least one element forms a good pair with infinitely many later elements. 4. An obvious approach is to try to imitate the proof of Lemma 12.1.3 for \( { \leq }^{\prime } \) ; if it fails, what is the reason? Alternatively, you might try to modify the injective map produced by Lemma 12.1.3 into an order-preserving one, without losing the property of \( a \leq f\left( a\right) \) for all \( a \) . 5. \( {}^{ - } \) This is an exercise in precision: ’easy to see’ is not a proof... 6. The trees in any bad sequence must get arbitrarily large. We are thus looking for trees \( T,{T}^{\prime } \) such that \( \left| T\right| < \left| {T}^{\prime }\right| \) but \( T \nleq {T}^{\prime } \) . Consider some simple examples, and iterate one to a bad sequence. 7. Does the original proof ever map the root of a tree to an ordinary vertex of another tree? 8. Can you extend a given graph \( G \) to another graph from which \( G \) can be obtained by deletion but not by contraction? Can you iterate this to build an infinite antichain? 9. \( {}^{ + } \) Can the graphs \( G \) in a bad sequence have arbitrarily many independent edges? If not, they have bounded-size subsets of vertices that cover all their edges. (Why?) Consider a subsequence where these vertex sets all induce the same graph, and find a good subsequence therein. 10. \( {}^{ + } \) When we try to embed a graph \( {TG} \) in another graph \( H \), the branch vertices of the \( {TG} \) can be mapped only to vertices of at least the same degree. Extend a suitable graph \( G \) to a similar graph \( H \) that does not contain \( G \) as a topological minor because these vertices are inconveniently positioned. Then iterate this example to obtain an infinite antichain. 11. \( {}^{ + } \) It is. One possible proof uses normal spanning trees with labels, and imitates the proof of Kruskal's theorem. 12. \( {}^{ - } \) The point about the ’subtrees’ is that they are connected. Recall our convention that connected graphs are non-empty. 13. \( {}^{ - } \) Start wi

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\( {}^{ + } \) Can the graphs \( G \) in a bad sequence have arbitrarily many independent edges? If not, they have bounded-size subsets of vertices that cover all their edges. (Why?) Consider a subsequence where these vertex sets all induce the same graph, and find a good subsequence therein. 10. \( {}^{ + } \) When we try to embed a graph \( {TG} \) in another graph \( H \), the branch vertices of the \( {TG} \) can be mapped only to vertices of at least the same degree. Extend a suitable graph \( G \) to a similar graph \( H \) that does not contain \( G \) as a topological minor because these vertices are inconveniently positioned. Then iterate this example to obtain an infinite antichain. 11. \( {}^{ + } \) It is. One possible proof uses normal spanning trees with labels, and imitates the proof of Kruskal's theorem. 12. \( {}^{ - } \) The point about the ’subtrees’ is that they are connected. Recall our convention that connected graphs are non-empty. 13. \( {}^{ - } \) Start with any tree-decomposition of least width and modify it in steps. 14. Why are there no cycles of tree-width 1 ? 15. For the forward implication, apply Corollary 1.5.2. For the converse, use induction on \( n \) . 16. To prove (T2), consider the edge \( e \) of Figure 12.3.1. Checking (T3) is easy. 17. For the first question, recall Proposition 12.3.6. For the second, try to modify a tree-decomposition of \( G \) into one of the \( {TG} \) without increasing its width. 18. \( {}^{ + } \) Use a normal spanning tree \( T \) as the decomposition tree, and let \( {t}_{1},\ldots ,{t}_{n} \) be an enumeration of \( V\left( T\right) \) such that \( {t}_{1} \) is the root and all the sets \( \left\{ {{t}_{1},\ldots ,{t}_{i}}\right\} \) are connected in \( T \) . Define the parts \( {V}_{t} \) inductively for \( t = {t}_{1},\ldots ,{t}_{n} \) so as to satisfy the condition in Exercise 15. 19. For (i), translate the compatibility condition to a similar condition on the components of \( T - e \) for the two choices of \( e \) . For (ii), either find an ingenious way to define the \( {V}_{t} \) directly, or apply induction on \( \left| \mathcal{S}\right| \) and delete from \( \mathcal{S} \) a separation \( \{ A, B\} \) with \( A \) minimal. In the tree-decomposition corresponding to \( \mathcal{S} \smallsetminus \{ \{ A, B\} \} \), find the part to which the new part should be joined by orienting the tree edges as in the proof of Lemma 12.3.4. 20. \( {}^{ + } \) For the first statement, let \( H = {H}_{t} \) be a torso that is not 3-connected. Show that there exists a cycle \( C = {v}_{1}\ldots {v}_{k}{v}_{1} \) with \( V\left( C\right) \subseteq V\left( H\right) \) (but not necessarily \( C \subseteq H \) ) such that, for all \( u, v, x, y \in V\left( C\right) \), the vertices \( u \) and \( v \) separate \( x \) from \( y \) in \( C \) if and only if they do so in \( H \) . Choose \( C \) maximal with respect to subdivision, and show that \( H = C \) . For the second statement, build the graph up inductively from the torsos of its tree-decomposition, chosen in an order that keeps the partial decomposition tree connected. 21. Modify the proof given in the text that the \( k \times k \) grid has tree-width at least \( k - 1 \) . 22. Existence was shown in Theorem 12.3.9; the task is to show uniqueness. 23. \( {}^{ + } \) Work out an explicit description of the sets \( {W}_{t}^{\prime } \) similar to the definition of the \( {W}_{t} \), and compare the two. 24. Induction. 25. Induction. 26. Use a result from Chapter 7.3. And don’t despair at a subgraph of \( W \) ! 27. \( {}^{ + } \) Show that the parts are precisely the maximal irreducible induced subgraphs of \( G \) . 28. Exercise 12. 29. For the forward implication, interpret the subpaths of the decomposition path as intervals. Which subpath corresponds naturally to a given vertex of \( G \) ? 30. Follow the proof of Corollary 12.3.12. 31. \( {}^{ + } \) They do. To prove it, show first that every connected graph \( G \) contains a path whose deletion decreases the path-width of \( G \) . Then apply induction on a suitable set of trees, deleting a suitable path in the induction step. 32. \( {}^{ - } \) Compare \( {\mathcal{K}}_{\mathcal{P}} \) with its analogue for the stronger notion. 33. To answer the first part, construct for each forbidden minor \( X \) a finite set of graphs whose exclusion as topological minors is equivalent to forbidding \( X \) as a minor. For the second part you may use Exercise 10. 34. \( {}^{ - } \) Find the required paths one by one. 35. \( {}^{ + } \) One direction is just a weakening of Lemma 12.4.5. For the other, imitate the proof of Lemma 12.3.4. 36. \( {}^{ + } \) Let \( X \) be an externally \( \ell \) -connected set of \( h \) vertices in a graph \( G \), where \( h \) and \( \ell \) are large. Consider a small separator \( S \) in \( G \) : clearly, most of \( X \) will lie in the same component of \( G - S \) . Try to make these ’large’ components, perhaps together with their separators \( S \), into the desired connected vertex sets. 37. A tangle of order \( k \) is a way of ’directing’ the separations of order \( < k \) . Direct them towards the set that Exercise 35 provides as a 'certificate' for large tree-width. 38. How much harder does it get to cover all the \( {MX} \) in \( G \) when \( \mathcal{X} \) and the graphs \( X \in \mathcal{X} \) get larger? How does the problem change if we replace \( \mathcal{X} \) by the set of its minor-minimal elements? 39. \( {}^{ + } \) Let \( S \) be a surface in which \( H \) can be embedded. You may use the fact that the number of copies of \( H \) that can be disjointly embedded in \( S \) is bounded by some number \( n \in \mathbb{N} \) . To show that \( f \) cannot be defined for \( k > n \), consider a candidate \( \ell \in \mathbb{N} \) for \( f\left( k\right) \) and extend a fixed drawing of \( H \) on \( S \) to a graph \( {H}^{\prime } \) on \( S \) that, after deleting any \( \ell \) vertices, still has an \( H \) minor. 40. \( {}^{ + } \) Find a counterexample. 41. \( {}^{ + } \) For an example showing that non-trivial tree-decompositions are necessary, use Exercise 31 and the fact that no surface can accommodate unboundedly many disjoint copies of \( {K}^{5} \) . For the remaining examples, work with modifications of large grids or grid-like graphs on other surfaces than the sphere. 42. Consult Chapter 7.2 for substructures to be found in graphs of large chromatic number. 43. \( {K}^{5} \) . 44. Derive the minor theorem first for connected graphs. 45. Use the separation properties of normal spanning trees proved in Chapter 1.5. If desired, you may use any exercise from Chapter 8. 46. Choose suitable rays in \( H \) as branch sets and new edges to join them. 47. For the first question, consider in the \( \mathbb{Z} \times \mathbb{Z} \) grid concentric cycles and paths between them, and use the fact that the \( \mathbb{Z} \times \mathbb{N} \) grid is planar. 48. \( {}^{ + } \) The proof of the forward implication differs from the finite case in that we now have to construct the decomposition tree together with the parts. Try to do this inductively, starting with a maximal complete subgraph \( H \) as the first part. To extend the decomposition into a component \( C \) of \( G - H \), consider a vertex in \( C \) with as many neighbours in \( H \) as possible, and show that these include all the neighbours of \( C \) in \( H \) . 49. For (i), assume that every finite subgraph of \( G \) has a chordal supergraph of clique number at most \( k \), and show that so does \( G \) . For (ii), add edges to make \( G \) edge-maximal with the property that every finite subgraph has tree-width at most \( k \) . Show that this supergraph of \( G \) must be chordal. 50. Planarity. You may use any exercise in Chapter 8. ## Index Page numbers in italics refer to definitions; in the case of author names, they refer to theorems due to that author. The alphabetical order ignores letters that stand as variables; for example, ’ \( k \) -chromatic’ is listed under the letter \( \mathrm{c} \) . above, 15 abstract dual, 105-106, 108 graph, 3, 83, 86, 92, 302 acyclic, 13-14, 48, 134 adhesion, 340, 341 adjacency matrix, 28, 32 adjacent, 3 Aharoni, R., 217, 223, 225, 226, 245, 247, 248 Ahuja, R.K., 161 algebraic colouring theory, 137 flow theory, 144-159, 161 graph theory, 23-28, 32 planarity criteria, 101-102 algorithmic graph theory, 161, 349, \( {355} - {356} \) almost, 302, 312-313 Alon, N., 10, 32, 122, 137-138, 314 alternating path, 34, 224 walk, 64 Andreae, Th., 207, 245, 246 antichain, 51, 53, 241, 316, 388, 389 antihole, 138 apex vertices, 340, 353 Appel, K., 137 arboricity, \( {46} - {49},{115},{190},{235},{250} \) arc, 84, 229, 243, 247, 248, 361, 385 -component, 229, 243 -connected, 229, 243, 248 Archdeacon, D., 355 Arnborg, S., 355 articulation point, see cutvertex at, 2 augmenting path for matching, 34, 51, 224, 241, 371 for network flow, 143, 160 automorphism, 3, 31, 215, 239, 374 average degree, 5 of bipartite planar graph, 376 bounded, 273 and choice number, 122 and chromatic number, 117, 122, 169, 172, 190 and connectivity, 12 forcing minors, 163, 170-171, 191, 192–194 forcing topological minors, 70, 169- 170 and girth, 8, 9-10, 301 and list colouring, 122 and minimum degree, 5-6 and number of edges, 5 and Ramsey numbers, 273 and regularity lemma, 176, 191 back-and-forth technique, 213-214, 383 bad sequence, 316, 354 balanced, 308 Bauer, 291 Behzad, M., 138 Bellenbaum, P., 355 below, 15 Berge, C., 128 Berger, E., 217, 247 between, 6, 84 Biggs, N.L., 32 binary tree, 203, 238 bipartite graphs, 17-18, 31, 107, 111, 127 edge colouring of, \( {119},{135},{136} \) flow number of cubic, 150 forced as subgraph, 169, 183 list-chromatic index of, 125-126, 138 matching in, 34–39, 222–224 in Ramsey theory

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

vertex at, 2 augmenting path for matching, 34, 51, 224, 241, 371 for network flow, 143, 160 automorphism, 3, 31, 215, 239, 374 average degree, 5 of bipartite planar graph, 376 bounded, 273 and choice number, 122 and chromatic number, 117, 122, 169, 172, 190 and connectivity, 12 forcing minors, 163, 170-171, 191, 192–194 forcing topological minors, 70, 169- 170 and girth, 8, 9-10, 301 and list colouring, 122 and minimum degree, 5-6 and number of edges, 5 and Ramsey numbers, 273 and regularity lemma, 176, 191 back-and-forth technique, 213-214, 383 bad sequence, 316, 354 balanced, 308 Bauer, 291 Behzad, M., 138 Bellenbaum, P., 355 below, 15 Berge, C., 128 Berger, E., 217, 247 between, 6, 84 Biggs, N.L., 32 binary tree, 203, 238 bipartite graphs, 17-18, 31, 107, 111, 127 edge colouring of, \( {119},{135},{136} \) flow number of cubic, 150 forced as subgraph, 169, 183 list-chromatic index of, 125-126, 138 matching in, 34–39, 222–224 in Ramsey theory, 263-264, 272 Birkhoff, G.D., 137 block, 55, 108, 372 graph, 56, 78, 372 Böhme, T., 81, 193 Bollobás, B., 54, 80, 192, 193, 245, 272, 291, 304, 305, 313, 314, 356 bond, \( {25},{31},{56},{104} - {106},{110},{238} \) -cycle duality, 104–106, 152–154 space, see cut space Bondy, J.A., 291 boundary circle, 361 of a face, \( {88} - {90},{107},{363} \) of a wave, 218 bounded subset of \( {\mathbb{R}}^{2},{86},{361} \) bounded graph conjecture, 238, 239, \( {244} - {245} \) bramble, 322-324, 351, 353, 355 number, 324 order of, 322 branch set, 19 in tree-decomposition, 325 vertex, 20 Brandt, S., 192 bridge, \( {11},{41},{141},{151},{156} - {157} \) to bridge, 281 Broersma, 291 Brooks, R.L., 115, 134 theorem, 115, 137 list colouring version, 137 Bruhn, H., 110, 247, 248, 278, 291 Burr, S.A., 272 Cameron, P.J., 246 capacity, 142 function, 141 cardinality, 357 Catlin, P.A., 193 Cayley, A., 137, 313 central face in grid, 342 vertex, \( 9,{342},{369} \) centre, 17 certificate, 126, 341, 356, 390 chain, 15, 51, 53, 241, 358, 360 Chebyshev inequality, 308, 388 Cherlin, G., 246 choice number, 121 and average degree, 122 of bipartite planar graphs, 135 of planar graphs, 122 \( k \) -choosable,121 chord, \( 8 \) chordal, 127-128, 136, 326, 352, 391 supergraph, 391 \( k \) -chromatic,111,134 chromatic index, 112, 119 of bipartite graphs, 119 vs. list-chromatic index, 121, 124 and maximum degree, 119-121 chromatic number, \( {111},{134},{155},{201} \) , 244, 353 and \( {K}^{r} \) -subgraphs, \( {116} - {117},{126},{226} \) of almost all graphs, 304 and average degree, \( {117},{122},{169} \) , 172, 190 vs. choice number, 121 and colouring number, 115 and connectivity, 116-117 constructions, 117-118, 134, 137 in extremal graph theory, 168 and flow number, 155 forcing minors, 172-175, 190, 191, \( {193} - {194} \) forcing short cycles, 117, 301 forcing subgraphs, 116-117, 238, 271 forcing a triangle, 135, 271 and girth, \( {117},{137},{175},{301} \) as a global phenomenon, 117, 126 and maximum degree, 115 and minimum degree, 115, 116 and number of edges, 114 chromatic polynomial, 134, 162 Chudnovsky, M., 128, 138 Chvátal, V., 256, 278, 279, 291 circle boundary circle, 361 in graph with ends, 106, 230, 231, 361 one/two-sided, 362 in surface, \( {348},{361},{362},{365} \) unit circle \( {S}^{1},{361} \) circuit, 23, 231, 242 circulation, 140-141, 153, 162 circumference, \( 8,{351} \) and connectivity, 79, 276 and minimum degree, 8 class 1 vs. class 2, 121 classification of surfaces, 361-362 clique number, 126-133, 263, 326 of a random graph, 296 threshold function, 312 closed under addition, 144, 232 under infinite sums, 235 under isomorphism, 3, 302, 327 wrt. minors, 135, 160, 245, 327, 341, 342, 349, 352 wrt. subgraphs, 126, 135 wrt. supergraphs, 126, 305 up or down, in tree-order, 15 walk, 10, 22 closure (of a set), 227 cocycle, see cut \( k \) -colourable,111,121,201,325 colour class, 111 colour-critical, see critically \( k \) -chromatic colouring, 111-138, 173, 201 algorithms, 114, 133 and flows, 152-155 number, 114, 134, 135, 245 plane graphs, 112-113, 152-155 in Ramsey theory, 253 total, 135, 138 3-colour theorem, see three colour thm. 4-colour theorem, see four colour thm. 5-colour theorem, see five colour thm. comb, 196, 242 modified, 240 star-comb lemma, 204 combinatorial isomorphism, 93, 94, 107, 108 set theory, 250, 272 Comfort, W.W., 250 compactness, 201, 227, 229, 242 proof technique, 200, 235-237, 238, 245 comparability graph, 127, 136 compatible separations, 351 complement of a bipartite graph, 127, 135 of a graph, 4 and perfection, 129, 376 of a property, 327, 341 complete bipartite graph, 17 graph, 3, 150 infinite graph, 197, 341 matching, see 1-factor minor, 97, 101, 169-175, 190, 191, \( {193} - {194},{340} - {341},{347} - {348} \) multipartite graph, 17, 167 part of path-decomposition, 352 part of tree-decomposition, 326 \( r \) -partite graph, \( {17} \) separator, 325, 352 subgraph, 117, 126-127, 163-167, 296, 312, 321 topological minor, 67-70, 81, 97, 101, \( {109},{169} - {170},{172},{175},{190},{194} \) complexity theory, 127, 341, 356 component, 11, 229, 361 connected, 10 arc-connected, 229, 243, 248 2-connected graphs, 55-57, 78, 89, 94, 270, 281 3-connected graphs, 57-62, 78, 89, \( {96},{97},{102},{269},{270} \) 4-connected graphs, 108, 270, 278 \( k \) -connected,11,12,67,79 externally, 329, 352 infinitely connected, 197, 237, 244 minimally connected, 14 minimally \( k \) -connected,80 semiconnected, 235-236 topologically, 229 and vertex enumeration, 10, 14 connectedness, 10, 14 connectivity, 11, 10-13, 55-81 and average degree, 12 and chromatic number, 116-117 and circumference, 79 and edge-connectivity, 12 external, 325, 329, 352, 353, 390 and girth, 237, 301 and Hamilton cycles, 277-278 in infinite graphs, 216-226 forcing minors, 354 and linkability, 70-71, 80, 81 and minimum degree, 12, 249 and plane duality, 108 and plane representation, 96 and Ramsey properties, 268-270 of a random graph, 303 via spanning trees, 46, 54 \( k \) -constructible, \( {117} - {118},{134},{137} \) contains, 3 continuum many, 357 contraction, 18-21 and 3-connectedness, 58-59 and minors, 18-21 in multigraphs, 28–30, 160 and tree-width, 320, 321 convex drawing, 99, 107, 109, 386 polygon, 271 core, 376 Corneil, D.G., 355 Cornuejols, G., 138 countable graph, 2 set, 357 countably infinite, 357 cover by antichains, 53 of a bramble, 322 by chains, 51 by edges, 136 by paths, 49-51, 223 by trees, 49, 106, 250 by vertices, 33, 34-35, 44-46, 322, 338 critical, 134 critically \( k \) -chromatic, \( {134},{375},{380} \) crosscap, 362, 364 cross-edges, 24, 46, 235 crosses in grid, 322 crown, 269-270 cube \( d \) -dimensional, \( {30},{313} \) of a graph, \( {G}^{3},{290} \) cubic graph, 5 connectivity of, 79 1-factor in, 41, 52 flow number of, \( {150},{151},{157},{161} \) , 162 multigraph, \( {44},{52},{157},{282} \) cuff, 339 Curran, S., 54 cut, 24 capacity of, 142, 143 -cycle duality, 104-106, 152-154 -edge, see bridge even/odd, 233, 243, 244, 249 flow across, 141 fundamental, 26, 32, 231, 243 minimal, \( {25},{31},{56},{104} \) in network, 142 space, \( {25} - {28},{31},{32},{101},{105},{249} \) cutvertex, 11, 55-56 cycle, \( 7 - 8 \) -bond duality, 104-106, 152-154 directed, 134, 135 disjoint cycles, 44-45 double cover conjecture, 157, 160 edge-disjoint cycles, 190, 240, 271 expected number, 298 facial, 101 fundamental, 26, 32, 382 Hamilton, 160, 275-291 infinite, 278, 289 induced, \( 8,{23},{59},{89},{102},{127},{128} \) , 243, 376, 380, 385 infinite, 106, 230–231, 249, 278 length, \( 8 \) long, \( 8,{30},{79},{134} \) in multigraphs, 29 non-separating, \( {59},{89},{102},{243},{385} \) odd, \( {17},{115},{128},{370},{376} \) with orientation, 152-154 short, 10, 117, 171–172, 299–301 space, \( {23} - {28},{31},{32},{59} - {62},{101} - {102} \) , 105, 107, 109, 232-235, 243, 244, 248, 249, 374 topological, 232-235, 248, 249 threshold function, 311, 313 cyclomatic number, 23 cylinder, 362 Czipszer, J., 249 Dean, N., 291 de Bruijn, N.G., 201, 245 degeneracy, see colouring number degree, 5 of an end, 204, 229, 231, 248 at a loop, 29 sequence, 278 deletion, 4 \( \Delta \) -system,271 dense graphs, 164, 167 linear order, 241 density edge density, 164 of pair of vertex sets, 176 upper density, 189 depth-first search tree, 16, 31 Deuber, W., 258, 273 diameter, \( 8 - 9,{312} \) and girth, 8 and radius, 9 Diestel, R., 110, 193, 216, 228, 233, 235, 244-250, 291, 340, 341, 355, 356 difference of graphs, 4, 86 digon, see double edge digraph, see directed graph Dilworth, R.P., 51, 53, 241, 372, 386 Dirac, G.A., 194, 276 directed cycle, 134, 135 edge, 28 graph, \( {28},{49} - {50},{124},{135},{246},{376} \) path, \( {49},{134},{375},{376} \) direction, 140 disc, 361 disconnected, 10 disjoint graphs, 3 dispersed, 239 distance, \( 8 \) dominated, 238, 249 double counting, 91, 109, 130-131, 298, 309 edge, 29, 103 ray, 196, 240, 250, 291 wheel, 269-270 down (-closure), 15 drawing, 2, 83, 92-96, 381 convex, 99, 109 straight-line, 99, 107 dual abstract, 105-106, 108 and connectivity, 108 plane, 103-105, 108 duality cycles and bonds, 26-28, 104-106, 152 flows and colourings, 152-155, 378 for infinite graphs, 106, 109, 110 of plane multigraphs, 103-106 tree-decompositions and brambles, 322 duplicating a vertex, 129, 166 edge, 2 crossing a partition, 24 directed, 28 double, 29 of a multigraph, 28 plane, 86 space, 23 topological, 226 \( X - Y \) edge,2 edge-chromatic number, see chromatic index edge colouring, 112, 119-121, 253, 259 and flow number, 151 and matchings, 135 \( \ell \) -edge-connected, 12 edge-connectivit

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

- {50},{124},{135},{246},{376} \) path, \( {49},{134},{375},{376} \) direction, 140 disc, 361 disconnected, 10 disjoint graphs, 3 dispersed, 239 distance, \( 8 \) dominated, 238, 249 double counting, 91, 109, 130-131, 298, 309 edge, 29, 103 ray, 196, 240, 250, 291 wheel, 269-270 down (-closure), 15 drawing, 2, 83, 92-96, 381 convex, 99, 109 straight-line, 99, 107 dual abstract, 105-106, 108 and connectivity, 108 plane, 103-105, 108 duality cycles and bonds, 26-28, 104-106, 152 flows and colourings, 152-155, 378 for infinite graphs, 106, 109, 110 of plane multigraphs, 103-106 tree-decompositions and brambles, 322 duplicating a vertex, 129, 166 edge, 2 crossing a partition, 24 directed, 28 double, 29 of a multigraph, 28 plane, 86 space, 23 topological, 226 \( X - Y \) edge,2 edge-chromatic number, see chromatic index edge colouring, 112, 119-121, 253, 259 and flow number, 151 and matchings, 135 \( \ell \) -edge-connected, 12 edge-connectivity, 12, 46, 67, 79, 134, 150, 197 edge contraction, 18 and 3-connectedness, 58 vs. minors, 19 in multigraph, 29 edge cover, 136 edge density, 5, 6, 164 and average degree, 5 forcing minors, 170 forcing path linkages, 71-77 forcing subgraphs, 164-169 forcing topological minors, 70, 169 and regularity lemma, 176, 191 edge-disjoint spanning trees, \( {46} - {49},{52} \) , 197 edge-maximal, 4 vs. extremal, 165, 173 without \( M{K}^{5},{174} \) without \( T{K}_{3,3},{191} \) without \( T{K}^{4},{173} \) without \( T{K}^{5}, T{K}_{3,3},{100} \) edge space, 23, 31, 101, 232 Edmonds, J., 53, 225, 356 embedding of bipartite graphs, 263-265 of graphs, 21 \( k \) -near embedding, \( {340} \) in the plane, 92, 95-110 in \( {S}^{2},{85} - {86},{93} \) self-embedding, 349 in surface, \( {91},{109},{341} - {349},{353},{356} \) , 363 empty graph, 2, 11 end degree, 204, 229, 231, 248 in subspaces, 229, 231, 248-249 of edge, 2, 28 -faithful spanning tree, 242 of graph, 49, 106, 195, 202-203, 204- 212, 226-244, 248-249 of path, 6 space, 226-237, 242 thick/thin, 208–212, 238 of topological space, 242 endpoints of arc, 84, 229 endvertex, 2, 28 terminal vertex, 28 enumeration, 357 equivalence in definition of an end, 202, 242 of graph invariants, 190 of graph properties, 270 of planar embeddings, 92-96, 106, 107 of points in topological space, 84, 361 in quasi-order, 350 Erdős, P., 45, 53, 117, 137, 167, 169, 185, 192, 193, 194, 201, 213, 216, 217, 244, 245, 246-247, 249, 250, 258, 271, 272, 273, 277, 291, 293- 294, 296, 299-301, 306, 308, 314, 387 Erdős-Menger conjecture, 217, 247 Erdős-Pósa property, 44, 52, 338-339, 353 Erdős-Pósa theorem, 45, 53 edge version, 190, 271 generalization, 338-339 Erdős-Sós conjecture, 169, 189-190, 193 Erdős-Stone theorem, 164, 167-168, 186-187, 193 Euler, L., 22, 32, 91 characteristic, 363 formula, 91-92, 106, 363, 376 genus, 343, 363-366 tour, 22, 244, 378, 385 Eulerian graph, 22 infinite, 233, 244, 248, 249-250 even degree, 22, 39 graph, \( {150},{151},{161},{248} \) event, 295 evolution of random graphs, 305, 313, 314 exceptional set, 176 excluded minors, see forbidden minors existence proof, probabilistic, 137, 293, 297, 299-301 expanding a vertex, 129 expectation, 297–298, 307 exterior face, see outer face external connectivity, 329, 352, 353 extremal bipartite graph, 189 vs. edge-maximal, 164-165, 173 graph theory, 163-194, 248-249 graph, 164-166 without \( M{K}^{5},{174} \) without \( T{K}_{3,3},{191} \) without \( T{K}^{4},{173} \) face, 86, 363 central face, 342 of hexagonal grid, 342 facial cycle, 101 factor, 33 1-factor, 33-43, 52, 216-226, 238, 241 1-factor theorem, \( {39},{41},{52},{53},{80} \) , \( {81},{225},{247} \) 2-factor, 39 \( k \) -factor, 33 factor-critical, 41, 225, 242, 371, 384 Fajtlowicz, S., 193 fan, 66, 238 -version of Menger's theorem, 66, 238 finite adhesion, 340, 341 graph, 2 set, 357 tree-width, 341 finite intersection property, 201 first order sentence, 303, 314 first point on frontier, 84 five colour theorem, 112, 137, 157 list version, 122, 138 five-flow conjecture, 156, 157, 162 Fleischner, H., 281, 289, 291, 387 flow, 139-162, 141-142 2-flow, 149 3-flow, 150, 157, 161 4-flow, 150-151, 156-157, 160, 161, 162 6-flow theorem, 157-159, 161, 162 \( k \) -flow, \( {147} - {151},{156} - {159},{160},{161} \) , 162 H-flow, 144-149, 160 -colouring duality, 152-155, 378 conjectures, 156-157, 161, 162 group-valued, \( {144} - {149},{160},{161} - {162} \) integral, 142, 144 network flow, \( {141} - {144},{160},{161},{378} \) number, \( {147} - {151},{156},{160},{161} \) in plane graphs, 152-155 polynomial, 146, 149, 162 total value of, 142 forbidden minors and chromatic number, 172-175 expressed by, 327, 340-349 in infinite graphs, \( {216},{244},{245},{340} \) 341 minimal set of, 341, 352, 355 planar, 328 and tree-width, 327-341 forcibly hamiltonian, see hamiltonian sequence forcing \( M{K}^{r},{169} - {175},{192} - {194},{340},{353} \) \( M{K}^{{\aleph }_{0}},{341},{354} \) \( T{K}^{5},{174},{193} \) \( T{K}^{r},{70},{169} - {170},{172},{175},{193} - {194} \) edge-disjoint spanning trees, 46 Hamilton cycles, 276-278, 281, 289 high connectivity, 12 induced trees, 169 large chromatic number, 117-118 linkability, 70-72, 81 long cycles, \( 8,{30},{79},{134},{275} - {291} \) long paths, 8, 30 minor with large minimum degree, 171, 193 short cycles, 10, 171-172, 175, 301 subgraph, 15, 163-169, 175-194 tree, \( {15},{169} \) triangle, 135, 271 Ford, L.R. Jr., 143, 161 forest, 13, 173, 327 minor, 355 partitions, 48-49, 53, 250 plane, 88, 106 topological, 250 tree-width of, 327, 351 four colour problem, 137, 193 four colour theorem, 112, 157, 161, 172, 174, 191, 278, 290 history, 137 four-flow conjecture, 156-157 Fraïssé, R., 246 Frank, A., 80, 161 Freudenthal, H., 248 compactification, 227, 248 ends, 242 Frobenius, F.G, 53 from...to, 6 frontier, 84, 361 Fulkerson, D.R., 122, 143, 161 fundamental circuit, 231, 233, 243 cocycle, 26, 32 cut, 26, 32, 231, 243 cycle, 26, 32 Gale, D., 38 Gallai, T., 32, 43, 50, 52, 53, 54, 81, 192, 238, 249 Gallai-Edmonds matching theorem, 41- \( {43},{53},{225},{247} \) Galvin, F., 125, 138 Gasparian, G.S., 129, 138 Geelen, J., 356 generated, 233 genus and colouring, 137 Euler genus, 343, 363-366 of a graph, 106, 353 orientable, 353 of a surface, 348 geometric dual, see plane dual Georgakopoulos, A., 248 Gibbons, A., 161 Gilmore, P.C., 136 girth, \( 8 \) and average degree, 9-10, 301 and chromatic number, 117, 137, \( {299} - {301} \) and connectivity, 81, 237, 301 and diameter, 8 and minimum degree, \( 8,{10},{30},{171} \) , 301 and minors, 170-172, 191, 193 and planarity, 106, 237 and topological minors, 172, 175 Godsil, C., 32 Golumbic, M.C., 138 good characterization, 341, 356 pair, 316, 347 sequence, 316 Gorbunov, K.Yu., 355 Göring, F., 81 Graham, R.L., 272 graph, 2-4, 28, 30 homogeneous, 215, 240, 246 invariant, 3, 30, 190, 297 minor theorem, 315, 341-348, 342, \( {349},{354},{355} \) for trees, 317-318 partition, 48 plane, 86–92, 103–106, 112–113, 122– 124, 152-155 process, 314 property, 3, 212, 270, 302, 312, 327, 342,356 simple, 30 universal, 212-216, 213, 240, 246 graphic sequence, see degree sequence graph-theoretical isomorphism, 93-94 greedy algorithm, 114, 124, 133 grid, 107, 208, 322 canonical subgrid, 342 hexagonal grid, 208, 209, 342-346 minor, 240, 324, 328-338, 354 theorem, 328 tree-width of, 324, 351, 354 Grötzsch, H., 113, 137, 157, 161 group-valued flow, \( {144} - {149},{160},{161} - \) 162 Grünwald, T., see Gallai Gusfield, D., 53 Guthrie, F., 137 Gyárfás, A., 169, 190, 194 Hadwiger, H., 172, 193 conjecture, 172-175, 191, 193 Hajnal, A., 244, 245, 249, 250, 258, 272, 273 Hajós, G., 118, 137, 175 conjecture, 175, 193 construction, 117-118 Haken, W., 137 Halin, R., 80, 206, 208, 244, 245-246, 354-355, 356 Hall, P., 36, 51, 53, 224 Hamilton, W.R., 290 Hamilton circle, 278, 289, 291 Hamilton cycle, 275–291 in \( {G}^{2},{281} - {289} \) in \( {G}^{3},{290} \) in almost all graphs, 305 and degree sequence, 278-281, 289 and the four colour theorem, 278 and 4-flows, 160, 278 in infinite graph, see Hamilton circle and minimum degree, 276 in planar graphs, 278 power of, 289 sufficient conditions, 275-281 Hamilton path, 275, 280-281, 289, 290 hamiltonian graph, 275 sequence, 279 handle, 362, 364 Harant, J., 81 head, see terminal vertex Heawood, P.J., 137, 161 Heesch, H., 137 height, 15 hexagonal grid, 208, 209, 342-346 Higman, D.G., 316, 354 Hoffman, A.J., 136 hole, 138 Holz, M., 247 homogeneous graphs, 215, 240, 246 Hoory, S., 10, 32 Huck, A., 244 hypergraph, 28 incidence, 2 encoding of planar embedding, see combinatorial isomorphism map, 29 matrix, 27 incident, 2, 88 incomparability graph, 242 increasing property, 305, 313 independence number, 126-133 and connectivity, 276-277 and covers, 50, 52 and Hamilton cycles, 276-277 and long cycles, 134 and perfection, 132 of random graph, 296, 312 independent edges, 3, 33-43, 52 events, 295 paths, 7, 66–67, 677–69, 370 vertices, 3, 50, 124, 296 indicator random variable, 298, 387 induced subgraph, 3-4, 68, 126, 128, 132, 376 of almost all graphs, 302, 313 cycle, \( 8,{23},{31},{59},{89},{102},{127},{128} \) , 249, 376, 380, 385 of all imperfect graphs, 129, 135 of all large connected graphs, 268 in Ramsey theory, 252, 258-268, 271 in random graph, 296, 313 tree, \( {169},{190} \) induction transfinite, 198–199, 359 Zorn's Lemma, 198, 237, 360 inductive ordering, 199 infinite graphs, \( 2,{19},{31},{51},{110},{189},{195} - \) 250, 253, 278, 289, 291, 305-306, 340-341, 349, 354, 356 sequence of steps, 197, 206 set, \( {357} \) basic properties, 197-198 infinitely conn

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

88 incomparability graph, 242 increasing property, 305, 313 independence number, 126-133 and connectivity, 276-277 and covers, 50, 52 and Hamilton cycles, 276-277 and long cycles, 134 and perfection, 132 of random graph, 296, 312 independent edges, 3, 33-43, 52 events, 295 paths, 7, 66–67, 677–69, 370 vertices, 3, 50, 124, 296 indicator random variable, 298, 387 induced subgraph, 3-4, 68, 126, 128, 132, 376 of almost all graphs, 302, 313 cycle, \( 8,{23},{31},{59},{89},{102},{127},{128} \) , 249, 376, 380, 385 of all imperfect graphs, 129, 135 of all large connected graphs, 268 in Ramsey theory, 252, 258-268, 271 in random graph, 296, 313 tree, \( {169},{190} \) induction transfinite, 198–199, 359 Zorn's Lemma, 198, 237, 360 inductive ordering, 199 infinite graphs, \( 2,{19},{31},{51},{110},{189},{195} - \) 250, 253, 278, 289, 291, 305-306, 340-341, 349, 354, 356 sequence of steps, 197, 206 set, \( {357} \) basic properties, 197-198 infinitely connected, 197, 237, 244 infinity lemma, 200, 245, 383 initial segment, 358 vertex, 28 inner face, 86 point, 226 vertex, 6 integral flow, 142, 144 function, 142 interior of an arc, 84 of a path, \( \overset{ \circ }{P},6 - 7 \) internally disjoint, see independent intersection, 3 graph, 352 interval graph, 127, 136, 352 into, 319 invariant, 3 irreducible graph, 352 Irving, R.W., 53 isolated vertex, 5, 313 isomorphic, 3 isomorphism, 3 of plane graphs, 92-96 isthmus, see bridge Itai, A., 54 Jaeger, F., 162 Janson, S., 313 Jensen, T.R., 136, 162, 355 Johnson, D., 356 join, 2 Jónsson, B., 246 Jordan, C., 84, 86 Jordan Curve Theorem, 84, 109 Jung, H.A., 70, 194, 205, 239, 245 Kahn, J., 138 Karoński, M., 314 Kawarabayashi, K., 193 Kelmans, A.K., 102, 109-110 Kempe, A.B., 137, 290 kernel of directed graph, 124, 135 of incidence matrix, 27 Kirchhoff's law, 139, 140 Klein four-group, 151 Kleitman, D.J., 137 knotless graph, 349 knot theory, 162 Kochol, M., 149, 162 Kohayakawa, Y., 194 Kollár, J., 192 Komlós, J., 192, 194, 272, 289, 291 König, D., 35, 53, 119, 200, 245 duality theorem, \( {35},{49},{51},{52},{63} \) , 127, 136, 223 infinity lemma, 200, 245 Königsberg bridges, 21 Korman, V., 226 Kostochka, A.V., 170, 192, 273 Kriesell, 53 Kruskal, J.A., 317, 354, 389 Kühn, D., 81, 172, 175, 193, 194, 216, 233, 246-250 Kuratowski, C., 96-101, 109, 238, 249, 356 -theorem for higher surfaces, 342 -type characterization, 107, 270, 341- 342, 355-356 Kuratowski set of graphs, 341–342, 355 of graph properties, 270 Lachlan, A.H., 215, 246 large wave, 218 Larman, D.G., 70 Latin square, 135 Laviolette, F., 250 Leader, I.B., 245, 246 leaf, 13, 15, 31, 204 lean tree-decomposition, 325 Lee, O., 54 length of a cycle, \( 8 \) of a path, 6, 8 of a walk, 10 level, 15 limit, 199-200, 358 wave, 218 line (edge), 2 graph, 4, 112, 136, 191 segment, 84 linear algebra, 23–28, 59–61, 101–102, 132 decomposition, 339-340 programming, 161 Linial, N., 10, 32 linkable, 219 linked by an arc, \( {84} \) by a path, 6 \( k \) -linked, \( {69} - {77},{80},{81},{170} \) vs. \( k \) -connected, \( {69} - {71},{80},{81} \) tree-decomposition, 325 vertices, 6, 84 list -chromatic index, 121, 124-126, 135, 138 -chromatic number, see choice num- ber colouring, 121–126, 137–138 bipartite graphs, 124-126, 135 Brooks's theorem, 137 conjecture, 124, 135, 138 \( k \) -list-colourable, see \( k \) -choosable Liu, X., 138 Lloyd, E.K., 32 locally finite, 196, 248, 249 logarithms, 1 loop, 28 Lovász, L., 53, 129, 132, 137, 138, 192 Luczak, T., 313, 314 MacLane, S., 101, 109-110 Mader, W., 12, 32, 67-69, 80, 81, 170, 190, 192, 193, 355 Magnanti, T.L., 161 Maharry, J., 193 Mani, P., 70 map colouring, \( {111} - {113},{133},{136},{152} \) Markov chain, 314 Markov's inequality, 297, 301, 307, 309 marriage theorem, \( {35} - {36},{39},{51},{53} \) , 223-224, 238, 371 stable, \( {38},{53},{126},{383} \) matchable, 41, 223 matching, 33-54 in bipartite graphs, 34-39, 127 and edge colouring, 135 in general graphs, 39-43 in infinite graphs, 222-226, 241-242, \( {247} - {248} \) partial, 224, 241 stable, \( {38},{51},{52},{126} \) of vertex set, 33 Máté, A., 250, 272 matroid theory, 54, 110, 356 max-flow min-cut theorem, 141, 143, 160, 161 maximal, 4 acyclic graph, 14 element, 358, 360 planar graph, \( {96},{101},{107},{109},{174} \) , 191, 374 plane graph, 90, 96 wave, 218 maximum degree, 5 bounded, 184, 256 and chromatic number, 115 and chromatic index, 119-121 and list-chromatic index, 126, 138 and radius, 9 and Ramsey numbers, 256-257 and total chromatic number, 135 Menger, K., 53, 62-67, 79, 81, 160, 206, 216-226, 241, 246-247 theorem of, 62-67, 79, 81, 160, 206- 207, 216, 217, 238, 246-247 \( k \) -mesh, \( {329} \) metrizable, 228, 242 Milgram, A.N., 50, 52, 53, 54 Milner, E.C., 245 minimal, 4 connected graph, 14 \( k \) -connected graph, \( {80} \) cut, \( {25},{31},{56},{104},{152} \) element, 358 non-planar graph, 107 separator, 78 set of forbidden minors, 341, 353, \( {355} - {356} \) minimum degree, 5 and average degree, 5 and choice number, 121-122 and chromatic number, 115, 116-117 and circumference, 8 and connectivity, 12, 80, 249 and edge-connectivity, 12 forcing Hamilton cycle, 276, 289 forcing long cycles, 8 forcing long paths, 8, 30 forcing short cycles, \( {10},{171} - {172},{175} \) , 301 forcing trees, 15 and girth, \( 8,9,{10},{170} - {172},{193},{301} \) and linkability, 71 minor, 18-21, 20, 169-172 \( {K}_{3,3},{109},{191} \) \( {K}^{4},{173},{327} \) \( {K}^{5},{174},{193},{352} \) \( {K}^{5} \) and \( {K}_{3,3},{96} - {101} \) \( {K}^{6},{175} \) \( {K}^{r},{170},{171},{172},{190},{191},{193} - {194} \) , 313, 340, 353, 354 \( {K}^{{\aleph }_{0}},{341},{354} \) of all large 3- or 4-connected graphs, \( {269} - {270} \) -closed graph property, 327, 341-349, 352 excluded, see forbidden forbidden, 172–175, 216, 244, 327– 349, 352, 354-356 forced, 171, 172, 169-175 incomplete, 192 infinite, 197, 207-208, 216, 240, 244, 245, 246, 248-249, 354, 356 of multigraph, 29 Petersen graph, 156 and planarity, 96-101, 107 proper, 349 relation, 20, 31, 207, 216, 240, 246, 270, 321, 342 theorem, 315, 341-349, 342, 354-355 proof, 342–348 for trees, 317-318 vs. topological minor, 20-21, 97 and WQO, 315-356 (see also topological minor) Möbius crown, 269-270 ladder, 174 strip, 362 Mohar, B., 109, 137, 193, 356 moment first, see Markov's inequality second, 306-312 monochromatic (in Ramsey theory) induced subgraph, 257-268 (vertex) set, 253-255 subgraph, 253, 255-257 Moore bound, 10, 32 multigraph, 28-30 cubic, \( {44},{52},{157},{282} \) list chromatic index of, 138 plane, 103 multiple edge, 28 multiplicity, 248 Murty, U.S.R., 291 Myers, J.S., 192 Nash-Williams, C.St.J.A., 46, 49, 53, 235, 244, 246, 247, 249-250, 291, 354 \( k \) -near embedding, \( {340} \) nearly planar, 340, 341 Negropontis, S., 250 neighbour of a set of vertices, 5 of a vertex, 3 Nešetřil, J., 272, 273 network, 141-144 theory, 161 Niedermeyer, F., 244, 248 node (vertex), 2 normal tree, \( {15} - {16},{31},{155},{160},{271},{389} \) in infinite graphs, 205, 228, 232, 239, 242, 245, 341, 356 ray, 205, 239, 384 nowhere dense, 49 zero, 144, 162 null, see empty obstruction to small tree-width, 322-324, 328- 329, 354, 355 octahedron, 12, 17, 355 odd component, 39, 238 cycle, \( {17},{115},{128},{135},{138},{370},{376} \) degree, 5, 290, 387 on, 2 one-factor theorem, \( \mathcal{{39}},{53},{81},{225} \) open Euler tour, 244 Oporowski, B., 269, 270, 273, 354 order of a bramble, 322 of a graph, 2 of a mesh or premesh, 329 partial, \( {15},{20},{31},{50} - {51},{53},{136} \) , 350, 357, 358, 360 quasi-, 316 of a separation, 11 tree-, 15, 31 type, 358 well-quasi-, 315-317, 342, 350, 354 ordinal, 358-359 orientable surface, 353 plane as, 153 orientation, \( {28},{124},{134},{161},{190},{376} \) cycle with, 152-153 oriented graph, 28, 289 Orlin, J.B., 161 Osthus, D., 81, 172, 175, 193, 194 outer face, 86, 93-94, 107 outerplanar, 107 Oxley, J.G., 93, 110, 250, 269, 270, 273 Oxtoby, J.C., 250 packing, 33, 44-49, 52, 235, 250 Palmer, E.M., 313 parallel edges, 29 parity, \( 5,{39},{42},{290} \) part of tree-decomposition, 319 partially ordered set, \( {50} - {51},{53},{241} \) , 358, 360 \( r \) -partite,17 partition, 1, 48, 253 pasting, 127, 173, 174, 191, 325, 352 path, 6-10, 196 \( a - b \) -path, \( 7,{66} \) \( A–B \) -path, \( 7,{62}–{67},{79},{216}–{223},{237} \) \( H \) -path, \( 7,{57},{67} - {69},{79},{80},{81} \) alternating, 34-35, 37, 63 between given pairs of vertices, 69-77 -connected, 248, 384 cover, 49-51, 50, 223, 372 -decomposition, 339, 352 directed, 49 disjoint paths, 50, 62–67, 69–77, 217- 222 edge-disjoint, 46, 66-67, 68-69 -hamiltonian sequence, 280-281 independent paths, \( 7,{66} - {67},{67} - {69} \) , \( {79},{80},{370} \) induced, 270 length, 6 linkage, 69-77, 81, 373 long, 8 -width, 352, 355 perfect, 126-133, 135-136, 137-138, 226 graph conjectures, 128 graph theorems, 128, 129, 135, 138 matching, see 1-factor strongly, 226, 242 weakly, 226, 242 Petersen, J., 39, 41 Petersen graph, 156-157 piecewise linear, 83 planar, \( {96} - {110},{112} - {113},{122},{216},{328} \) , 338, 341 embedding, 92, 96–110 nearly planar, 340, 341 planarity criteria Kelmans, 102 Kuratowski, 101 MacLane, 101 Tutte, 109 Whitney, 105 plane dual, 103 duality, 103-106, 108, 152-155 graph, 86-92 multigraph, 103-106, 108, 152-155 triangulation, 90, 91, 161, 325 Plummer, M.D., 53 Podewski, K.P., 247, 248 point (vertex), 2 pointwise greater, 279 Polat, N., 248 polygon, 84 polygonal arc, 84, 85 Pósa, L., 45, 53, 258, 273 power of a graph, 281 set, 357 predecessor, 358 preferences, 38, 51

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

hamiltonian sequence, 280-281 independent paths, \( 7,{66} - {67},{67} - {69} \) , \( {79},{80},{370} \) induced, 270 length, 6 linkage, 69-77, 81, 373 long, 8 -width, 352, 355 perfect, 126-133, 135-136, 137-138, 226 graph conjectures, 128 graph theorems, 128, 129, 135, 138 matching, see 1-factor strongly, 226, 242 weakly, 226, 242 Petersen, J., 39, 41 Petersen graph, 156-157 piecewise linear, 83 planar, \( {96} - {110},{112} - {113},{122},{216},{328} \) , 338, 341 embedding, 92, 96–110 nearly planar, 340, 341 planarity criteria Kelmans, 102 Kuratowski, 101 MacLane, 101 Tutte, 109 Whitney, 105 plane dual, 103 duality, 103-106, 108, 152-155 graph, 86-92 multigraph, 103-106, 108, 152-155 triangulation, 90, 91, 161, 325 Plummer, M.D., 53 Podewski, K.P., 247, 248 point (vertex), 2 pointwise greater, 279 Polat, N., 248 polygon, 84 polygonal arc, 84, 85 Pósa, L., 45, 53, 258, 273 power of a graph, 281 set, 357 predecessor, 358 preferences, 38, 51, 126 premesh, 329 Prikry, K., 245 probabilistic method, 293, 299-302, 314 projective plane, 355 proper minor, 349 separation, 11 subgraph, 3 wave, 218 property, 3, 270, 302 of almost all graphs, 302-306, 311- 312 increasing, 305 minor-closed, 327, 352 Proskurowski, A., 355 pseudo-random graph, 272 Pym, J.S., 223, 247 quasi-ordering, 315-317, 342, 350, 354 radius, \( 9 \) and diameter, 9, 30 and maximum degree, 9 Rado, R., 245, 246, 250, 272 graph, 214-215, 240, 241, 246, 306 Rado's selection lemma, 245 Ramsey, F.P., 252-255 Ramsey graph, 258 -minimal, 257-258 numbers, 253, 255, 271, 272-273, 296,314 Ramsey theory, 251-273 and connectivity, 268-270 induced, 258-268 infinite, 253–254, 271, 272 random graph, 170, 175, 255, 293-314, 295 evolution, 305, 311, 314 infinite, 305-306 process, 314 uniform model, 314 random variable, 297 indicator r.v., 298, 387 ray, \( {196},{200},{204},{206},{239},{240},{242} \) , 341 double, 196, 240, 250, 291 normal, 205, 239, 384 spanning, 291 recursive definition, 359-360 reducible configuration, 137 Reed, B.A., 53, 355 refining a partition, 1, 178-182 region, \( {84} - {86} \) on \( {S}^{2},{86} \) regular, \( 5,{37},{39},{135},{289} \) \( \epsilon \) -regular pair, 176, 191 partition, 176 regularity graph, 184 inflated, \( {R}_{s},{256} \) lemma, 164, 175-188, 176, 191, 193- 194, 272 Rényi, A., 213, 246, 306, 308, 314 Richardson, M., 135 Richter, B., 356 rigid-circuit, see chordal Riha, S., 291 ring, 342-343 Robertson, N., 53, 128, 137, 138, 162, 175, 193, 321, 328, 340, 341, 342, 354-355, 356 Rödl, V., 194, 256, 258, 272-273 Rónyai, L., 192 root, 15 rooted tree, 15, 317, 350 Rothschild, B.L., 272 Royle, G.F., 32 Ruciński, A., 313, 314 Salazar, G., 356 Sanders, D.P., 137 Sárközy, G.N., 289, 291 saturated, see edge-maximal Sauer, N., 246 Schelp, R.H., 210 Schoenflies, A.M., 86 Schrijver, A., 53, 80, 81, 138, 161 Schur, I., 271 Scott, A.D., 194, 246 second moment, 306-312, 307 self-minor conjecture, 349, 353, 354 semiconnected, 235-236 separate a graph, 11, 62, 66, 67 the plane, 84 separating circle, 362, 365 separation, 11 compatible, 351 order of, 11 and tree-decompositions, 320, 351, 353 separator, 11 sequential colouring, see greedy algo- rithm series-parallel, 191 set \( k \) -set,1 countable, 357 countably infinite, 357 finite, \( {357} \) infinite, 357 power set, 357 system, see hypergraph well-founded, 358 Seymour, P.D., 53, 128, 137, 138, 157, 162, 175, 193, 289, 291, 321, 322, 328, 340, 341, 342, 349, 354, 355, 356 Shapley, L.S., 38 Shelah, S., 244, 245, 246, 247 Shi, N., 246 shift-graph, 271 Simonovits, M., 53, 192, 194, 272 simple basis, 101, 109 graph, 30 simplicial tree-decomposition, 244, 325, 352,355 six-flow theorem, 157, 162 small wave, 218 snark, 157 planar, 157, 161, 278 Sós, V., 169, 189, 190, 192 spanned subgraph, 4 spanning ray, 291 subgraph, 4 trees, 14, 16 edge-disjoint, 46-49 end-faithful, 242 normal, 15-16, 31, 205, 228, 232, 239, 242, 245, 341, 356 number of, 313 topological, 49, 231-237, 242, 243, 250, 385 sparse graphs, 163, 169-172, 191, 194, \( {255} - {256},{273} \) Spencer, J.H., 272, 314 Sperner's lemma, 51 sphere \( {S}^{2},{86},{93} - {95},{361} \) spine, 196 Sprüssel, Ph., 32 square of a graph, 281-289, 290, 291 Latin, 135 stability number, see independence number stable marriage, 38, 53, 126, 383 matching, 38, 51, 52, 126 set, 3 standard basis, 23 subspace, 227, 231, 236, 243 star, 17, 190, 258, 270 centre of, 17 induced, 268 infinite, 204 -shape, 374 star-comb lemma, 204, 205 Steffens, K., 224, 247 Stein, M., 247, 248, 250 Steinitz, E., 109 stereographic projection, 85 Stillwell, J., 109 Stone, A.H., 167, 183 straight line segment, 84 strip neighbourhood, 88, 362 strong core, 376 strongly perfect, 226, 242 subcontraction, see minor subdividing vertex, 20 subdivision, 20 subgraph, 3 of all large \( k \) -connected graphs,268- 270 forced by edge density, 164-169, 175- \( {188},{189},{190},{191} \) of high connectivity, 12 induced, 3 of large minimum degree, 6, 115, 134 spanning, 4 successor, 358 Sudakov, B., 273 sum of edge sets, 23 of flows, 149 of thin families, 232 supergraph, 3 suppressing a vertex, 29 surface, 339, 342, 343, 361-367 surgery on, 364 surgery on surfaces, 364 capping, 364 cutting, 364 symmetric difference, 23, 34, 64 system of distinct representatives, 51 Szabó, T., 192 Szekeres, G., 271 Szemerédi, E., 176, 192, 194, 256, 272, 289, 291 see also regularity lemma tail of an edge, see initial vertex of a ray, 196, 237 Tait, P.G., 137, 290-291 tangle, 353, 355 Tarsi, M., 137 teeth, 196 terminal vertex, 28 thick/thin end, 208-212, 238 thin end, 208-212, 238 family, 232 sum, 232 Thomas, R., 53, 71, 81, 128, 137, 138, 162, 175, 193, 269, 270, 273, 291, 322, 325, 340, 341, 354, 355, 356 Thomason, A.G., 170, 192, 305 Thomassé, S., 246 Thomassen, C., 80, 109, 122, 137, 138, 171, 193, 244, 247, 291, 355, 356, 365 three colour theorem, 113 three-flow conjecture, 157 threshold function, 305-312, 313, 314 Toft, B., 136, 162 topological connectedness, 229, 236 cycle space, 232-235, 248, 249 edge, 226 end degree, 229 end space, 226-237, 242 Euler tour, 244 forest, 250 isomorphism, 93, 94, 104 spanning tree, 49, 231-237, 242, 243, 250, 385 topological minor, 20 \( {K}_{3,3},{92},{97},{100},{101},{109},{191} \) \( {K}^{4},{59},{173} - {174},{191},{327} \) \( {K}^{5},{92},{97},{100},{101},{109},{174},{193}, \) 352 \( {K}^{5} \) and \( {K}_{3,3},{92},{97},{100},{101},{107} \) , 109 \( {K}^{r},{70},{165},{169} - {172},{175},{190},{191} \) , 193–194, 252, 268, 340 \( {K}^{{\aleph }_{0}},{197},{205},{238},{241},{341},{354} \) of all large 2-connected graphs, 269 forced by average degree, 70, 169-172 forced by chromatic number, 175 forced by girth, 172, 175 induced, 170 as order relation, 20 vs. ordinary minor, 20, 97 and planarity, 92, 96-101 tree (induced), 169 and WQO of general graphs, 350 and WQO of trees, 317 torso, 339-341 total chromatic number, 135 total colouring, 135 conjecture, 135, 138 total value of a flow, 142 touching sets, 322 \( t \) -tough, \( {277} - {278},{290} \) toughness conjecture, 278, 289, 290, 291 tournament, 289 transfinite induction, 198-199, 359 transitive graph, 52 travelling salesman problem, 290 tree, 13-16 binary, 203, 238 cover, 46-49 as forced substructure, 15, 169, 190 level of, 15 normal, \( {15} - {16},{31},{155},{160},{389} \) infinite, 205, 228, 232, 239, 242, \( {245},{341},{356} \) -order, 15 -packing, \( {46} - {48},{52},{53},{235},{249},{250} \) path-width of, 352 spanning, 14, 16, 198, 205 topological, 49, 231-237, 242, 243, 250, 385 threshold function for, 312 well-quasi-ordering of trees, 317-318 tree-decomposition, 193, 319-326, 340, 341, 351, 354-355 induced on minors, 320 induced on subgraphs, 320 lean, 325 obstructions, 322-324, 328-329, 354, 355 part of, 319 simplicial, 325, 339, 352, 355 width of, 321 tree-packing theorem, 46, 235 tree-width, 321-341 and brambles, 322-324, 353, 355 compactness theorem, 354 duality theorem, 322-324 finite, 341 and forbidden minors, 327-341 of grid, 324, 351, 354 of a minor, 321 obstructions to small, 322-324, 328- 329, 354, 355 of a subdivision, 351 triangle, 3 triangulated, see chordal triangulation, see plane triangulation trivial graph, 2 Trotter, W.T., 256, 272 Turán, P., 165 theorem, 165, 192, 256 graph, 165-169, 192, 379 Tutte, W.T., 39, 46, 53, 57, 58, 59, 80, \( {102},{109},{144},{147},{155},{161} - {162}, \) 225, 235, 250, 278, 291 condition, 39-40 cycle basis theorem, 59, 249 decomposition of 2-connected graphs into 3-connected pieces, 57 1-factor theorem, 39, 53, 225 flow conjectures, 156-157, 162 planarity criterion, 102, 109 polynomial, 162 tree-packing theorem, \( {46},{53} - {54},{235} \) , 250 wheel theorem, 58-59, 80 Tychonoff's theorem, 201, 245, 381 ubiquitous, 207, 240, 246 conjecture, 207, 240, 246 unbalanced subgraph, 312, 313, 314 unfriendly partition conjecture, 202, 245 uniformity lemma, see regularity lemma union, 3 unit circle \( {S}^{1},{84},{361} \) universal graphs, 212-216, 213, 240, 246 unmatched, 33 up (-closure), 15 upper bound, 358 density, 189 Urquhart, A., 137 valency (degree), 5 value of a flow, 142 variance, 307 Veldman, H.J., 291 Vella, A., 249 vertex, 2 -chromatic number, 111 colouring, 111, 114-118 -connectivity, 11 cover, 34, 49-51 cut, see separator duplication, 166 expansion, 129 of a plane graph, 86 space, 23 -transitive, 52, 215, 239 Vince, A., 314 Vizing, V.G., 119, 137, 138, 376, 377, 380 Voigt, M., 137-138 vortex, 340, 353 Vušković, K., 138 Wagner, K., 101, 109, 174, 190, 191, 193, 354-355 'Wagner's Conjecture', see graph minor theorem Wagner graph, 174, 325-326, 352 walk, 10 alternating, 64 closed, 10 length, 10 wave, 217, 241 large, 218 limit, 218

1007_(GTM173)Graph.Theory,.Reinhard.Diestel.（图论）

, 80 Tychonoff's theorem, 201, 245, 381 ubiquitous, 207, 240, 246 conjecture, 207, 240, 246 unbalanced subgraph, 312, 313, 314 unfriendly partition conjecture, 202, 245 uniformity lemma, see regularity lemma union, 3 unit circle \( {S}^{1},{84},{361} \) universal graphs, 212-216, 213, 240, 246 unmatched, 33 up (-closure), 15 upper bound, 358 density, 189 Urquhart, A., 137 valency (degree), 5 value of a flow, 142 variance, 307 Veldman, H.J., 291 Vella, A., 249 vertex, 2 -chromatic number, 111 colouring, 111, 114-118 -connectivity, 11 cover, 34, 49-51 cut, see separator duplication, 166 expansion, 129 of a plane graph, 86 space, 23 -transitive, 52, 215, 239 Vince, A., 314 Vizing, V.G., 119, 137, 138, 376, 377, 380 Voigt, M., 137-138 vortex, 340, 353 Vušković, K., 138 Wagner, K., 101, 109, 174, 190, 191, 193, 354-355 'Wagner's Conjecture', see graph minor theorem Wagner graph, 174, 325-326, 352 walk, 10 alternating, 64 closed, 10 length, 10 wave, 217, 241 large, 218 limit, 218 maximal, 218 proper, 218 small, 218 weakly perfect, 226, 242 well-founded set, 358 well-ordering, 358, 386 theorem, 358 well-quasi-ordering, 316-356 Welsh, D.J.A., 162 wheel, 59, 270 theorem, 58-59, 80 Whitney, H., 81, 96, 105 width of tree-decomposition, 321 Wilson, R.J., 32 Winkler, P., 314 Wollan, P., 71, 81 Woodrow, R.E., 215, 246 Yu, X., 54, 291 Zehavi, A., 54 Zorn's lemma, 198, 237, 360 Zykov, A.A., 192 ## Symbol Index The entries in this index are divided into two groups. Entries involving only mathematical symbols (i.e. no letters except variables) are listed on the first page, grouped loosely by logical function. The entry ’[ ]’, for example, refers to the definition of induced subgraphs \( H\left\lbrack U\right\rbrack \) on page 4 as well as to the definition of face boundaries \( G\left\lbrack f\right\rbrack \) on page 88 . Entries involving fixed letters as constituent parts are listed on the second page, in typographical groups ordered alphabetically by those letters. Letters standing as variables are ignored in the ordering. \( \langle \;,\;\rangle \) 23 / 18, 19, 29 23 \( \overline{0},\overline{1},\overline{2},\ldots \;1 \) \( \leq \) 317,357 \[ {\left( n\right) }_{k},\ldots \] 298 久 \[ E\left( v\right) ,{E}^{\prime }\left( w\right) ,\ldots \] 2 \[ 4,{23},{144} \] \[ E\left( {X, Y}\right) ,{E}^{\prime }\left( {U, W}\right) ,\ldots \] 2 \[ \text{4, 86, 144} \] \[ \left( {e, x, y}\right) ,\left( {u, v}\right) ,\ldots \] 140, , 226 \( \overrightarrow{E},\overrightarrow{F},\overrightarrow{C},\ldots \;{140},{152},{154} \) \[ \overleftarrow{e},\overleftarrow{E},\overleftarrow{F},\ldots \] 140 \[ f\left( {X, Y}\right), g\left( {U, W}\right) ,\ldots \] 140 03, 152 \[ {G}^{2},{H}^{3},\ldots \] 281  \[ \bar{G},\bar{X},\overline{\mathcal{P}},\ldots 4,{140},{227},{327} \] \[ \left( {S,\overline{S}}\right) ,\ldots \] 2, 142, 226 \( {xy},{x}_{1}\ldots {x}_{k},\ldots \) 2, 7 \( {xP},{Px},{xPy},{xPyQz},\ldots \) \( \overset{ \circ }{P},\overset{ \circ }{x}Q,\overset{ \circ }{F},\ldots \) 7, 84, 226 \( \left\lbrack \;\right\rbrack < \omega \) 14 <table><tr><td>\( {\mathbb{F}}_{2} \)</td><td>23</td><td>\( \operatorname{col}\left( G\right) \)</td><td>114</td></tr><tr><td>N</td><td>1, 357</td><td>\( d\left( G\right) \)</td><td>5</td></tr><tr><td>\( {\mathbb{Z}}_{n} \)</td><td>1</td><td>\( d\left( v\right) \)</td><td>5</td></tr><tr><td colspan="2"></td><td>\( {d}^{ + }\left( v\right) \)</td><td>124</td></tr><tr><td>\( {\mathcal{C}}_{G} \)</td><td>39</td><td>\( d\left( {x, y}\right) \)</td><td>8</td></tr><tr><td>\( \mathcal{C}\left( G\right) \)</td><td>23, 232</td><td>\( d\left( {X, Y}\right) \)</td><td>176</td></tr><tr><td>\( {\mathcal{C}}^{ * }\left( G\right) \)</td><td>25, 249</td><td>\( \operatorname{diam}G \)</td><td>8</td></tr><tr><td>\( \mathcal{E}\left( G\right) \)</td><td>23</td><td>\( \operatorname{ex}\left( {n, H}\right) \)</td><td>165</td></tr><tr><td>\( \mathcal{G}\left( {n, p}\right) \)</td><td>294</td><td>\( {f}^{ * }\left( v\right) \)</td><td>103</td></tr><tr><td>\( {\mathcal{K}}_{\mathcal{P}},{\mathcal{K}}_{\mathcal{P}\left( S\right) } \)</td><td>341, 342</td><td>\( g\left( G\right) \)</td><td>8</td></tr><tr><td>\( {\mathcal{P}}_{H} \)</td><td>308</td><td>\( \bar{i} \)</td><td>1</td></tr><tr><td>\( {\mathcal{P}}_{i, j} \)</td><td>302</td><td>\( \operatorname{init}\left( e\right) \)</td><td>28</td></tr><tr><td>\( \mathcal{V}\left( G\right) \)</td><td>23</td><td>\( \log \) , \( \ln \) \( \operatorname{pw}\left( G\right) \)</td><td>1 352</td></tr><tr><td>\( {C}^{k}, C\left( {S,\omega }\right) ,{\widehat{C}}_{\epsilon }\left( {S,\omega }\right) \)</td><td>8, 227</td><td>\( q\left( G\right) \)</td><td>39</td></tr><tr><td>\( E\left( G\right) \)</td><td>2</td><td>\( \operatorname{rad}G \)</td><td>9</td></tr><tr><td>\( E\left( X\right) \)</td><td>297</td><td>\( {t}_{r - 1}\left( n\right) \)</td><td>165</td></tr><tr><td>\( F\left( G\right) \)</td><td>86</td><td>\( \operatorname{ter}\left( e\right) \)</td><td>28</td></tr><tr><td>\( {\operatorname{Forb}}_{ \preccurlyeq } \)</td><td>327</td><td>\( \operatorname{tw}\left( G\right) \)</td><td>321</td></tr><tr><td>\( G\left( {{H}_{1},{H}_{2}}\right) \)</td><td>259</td><td>\( {v}_{e},{v}_{xy},{v}_{U} \)</td><td>18, 19</td></tr><tr><td>\( {K}^{n} \)</td><td>3, 197</td><td>\( {v}^{ * }\left( f\right) \)</td><td>103</td></tr><tr><td>\( {K}_{{n}_{1},\ldots ,{n}_{r}} \)</td><td>17</td><td></td><td></td></tr><tr><td>\( {K}_{s}^{r} \)</td><td>17</td><td>\( \Delta \left( G\right) \)</td><td>5</td></tr><tr><td>\( L\left( G\right) \)</td><td>4</td><td></td><td></td></tr><tr><td>MX</td><td>19</td><td>\( \alpha \left( G\right) \)</td><td>126</td></tr><tr><td>\( N\left( v\right), N\left( U\right) \)</td><td>5</td><td>\( \delta \left( G\right) \)</td><td>5</td></tr><tr><td>\( {N}^{ + }\left( v\right) \)</td><td>124</td><td>\( \varepsilon \left( G\right) ,\varepsilon \left( S\right) \)</td><td>5, 363</td></tr><tr><td>\( P \)</td><td>295</td><td>\( \kappa \left( G\right) \)</td><td>11</td></tr><tr><td>\( {P}^{k} \)</td><td>6</td><td>\( {\kappa }_{G}\left( H\right) \)</td><td>68</td></tr><tr><td>\( {P}_{G} \)</td><td>134</td><td>\( \lambda \left( G\right) \)</td><td>12</td></tr><tr><td>\( R\left( H\right) \)</td><td>255</td><td>\( {\lambda }_{G}\left( H\right) \)</td><td>68</td></tr><tr><td>\( R\left( {{H}_{1},{H}_{2}}\right) \)</td><td>255</td><td>\( \mu \)</td><td>307</td></tr><tr><td>\( R\left( {k, c, r}\right) \)</td><td>255</td><td colspan="2">\( \pi : {S}^{2} \smallsetminus \{ \;\left( {0,0,1}\right) \;\} \rightarrow {\mathbb{R}}^{2}\;{85} \)</td></tr><tr><td>\( R\left( r\right) \)</td><td>253</td><td>\( {\sigma }_{k} : \mathbb{Z} \rightarrow {\mathbb{Z}}_{k} \)</td><td>147</td></tr><tr><td>\( {R}_{s} \)</td><td>184</td><td>\( {\sigma }^{2} \)</td><td>307</td></tr><tr><td>\( {S}^{n} \)</td><td>85, 361</td><td>\( \varphi \left( G\right) \)</td><td>147</td></tr><tr><td>\( {T}_{2} \)</td><td>203</td><td>\( \chi \)</td><td>111, 363</td></tr><tr><td>\( {TX} \)</td><td>20</td><td>\( {\chi }^{\prime }\left( G\right) \)</td><td>112</td></tr><tr><td>\( {T}^{r - 1}\left( n\right) \)</td><td>165</td><td>\( {\chi }^{\prime \prime }\left( G\right) \)</td><td>135</td></tr><tr><td>\( V\left( G\right) \)</td><td>2</td><td>\( \omega \left( G\right) ,\omega \)</td><td>126, 358</td></tr><tr><td></td><td></td><td>\( \Omega \left( G\right) \)</td><td>203</td></tr><tr><td>\( \operatorname{ch}\left( G\right) \)</td><td>121</td><td></td><td></td></tr><tr><td>\( {\operatorname{ch}}^{\prime }\left( G\right) \)</td><td>121</td><td>\( {\aleph }_{0},{\aleph }_{1} \)</td><td>357</td></tr></table> Reinhard Diestel received a PhD from the University of Cambridge, following research 1983-86 as a scholar of Trinity College under Béla Bollobás. He was a Fellow of St. John's College, Cambridge, from 1986 to 1990. Research appointments and scholarships have taken him to Bielefeld (Germany), Oxford and the US. He became a professor in Chemnitz in 1994 and has held a chair at Hamburg since 1999. Reinhard Diestel's main area of research is graph theory, including infinite graph theory. He has published numerous papers and a research monograph, Graph Decompositions (Oxford 1990).

1008_(GTM174)Foundations of Real and Abstract Analysis

# GraduateTexts inMathematics Douglas S. Bridges # Foundations of Real and Abstract Analysis Springer ## Graduate Texts in Mathematics 174 Editorial Board S. Axler F.W. Gehring K.A. Ribet Springer New York Berlin Heidelberg Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo ## Graduate Texts in Mathematics 1 TAKEUTI/ZARING. Introduction to Axiomatic Set Theory. 2nd ed. 2 Oxtoby. Measure and Category. 2nd ed. 3 Schaefer. Topological Vector Spaces. 4 Hilton/Stammbach. A Course in Homological Algebra. 2nd ed. 5 MAC LANE. Categories for the Working Mathematician. 6 Hughes/Piper. Projective Planes. 7 Serre. A Course in Arithmetic. 8 TAKEUTI/ZARING. Axiomatic Set Theory. 9 Humphreys. Introduction to Lie Algebras and Representation Theory. 10 COHEN. A Course in Simple Homotopy Theory. 11 Conway. Functions of One Complex Variable I. 2nd ed. 12 BEALS. Advanced Mathematical Analysis. 13 Anderson/Fuller. Rings and Categories of Modules. 2nd ed. 14 Golubitsky/Gunlemin. Stable Mappings and Their Singularities. 15 Berberlan. Lectures in Functional Analysis and Operator Theory. 16 WINTER. The Structure of Fields. 17 Rosenblatt. Random Processes. 2nd ed. 18 Halmos. Measure Theory. 19 Halmos. A Hilbert Space Problem Book. 2nd ed. 20 Husemoller. Fibre Bundles. 3rd ed. 21 Humphreys. Linear Algebraic Groups. 22 BARNES/MACK. An Algebraic Introduction to Mathematical Logic. 23 Greub. Linear Algebra. 4th ed. 24 Holmes. Geometric Functional Analysis and Its Applications. 25 Hewrrt/Stromberg. Real and Abstract Analysis. 26 Manes. Algebraic Theories. 27 Kelley. General Topology. 28 ZARISKI/SAMUEL. Commutative Algebra. Vol.I. 29 ZARISKI/SAMUEL. Commutative Algebra. Vol.II. 30 JACOBSON. Lectures in Abstract Algebra I. Basic Concepts. 31 JACOBSON. Lectures in Abstract Algebra II. Linear Algebra. 32 JACOBSON. Lectures in Abstract Algebra III. Theory of Fields and Galois Theory. 33 Hirsch. Differential Topology. 34 Spitzer. Principles of Random Walk. 2nd ed. 35 Wermer. Banach Algebras and Several Complex Variables. 2nd ed. 36 Kelley/Namioka et al. Linear Topological Spaces. 37 Monk. Mathematical Logic. 38 Grauert/Fritzsche. Several Complex Variables. 39 Arveson. An Invitation to \( {C}^{ * } \) -Algebras. 40 Kemeny/Snell/Knapp. Denumerable Markov Chains. 2nd ed. 41 APOSTOL. Modular Functions and Dirichlet Series in Number Theory. 2nd ed. 42 Serre. Linear Representations of Finite Groups. 43 Gillman/Jerison. Rings of Continuous Functions. 44 Kendig. Elementary Algebraic Geometry. 45 Loève. Probability Theory I. 4th ed. 46 Loève. Probability Theory II. 4th ed. 47 MOISE. Geometric Topology in Dimensions 2 and 3. 48 SACHS/WU. General Relativity for Mathematicians. 49 Gruenberg/Weir. Linear Geometry. 2nd ed. 50 Edwards. Fermat's Last Theorem. 51 KLINGENBERG. A Course in Differential Geometry. 52 Hartshorne. Algebraic Geometry. 53 Mann. A Course in Mathematical Logic. 54 Graver/Watkins. Combinatorics with Emphasis on the Theory of Graphs. 55 Brown/Pearcy. Introduction to Operator Theory I: Elements of Functional Analysis. 56 Massey. Algebraic Topology: An Introduction. 57 Crowell/Fox. Introduction to Knot Theory. 58 KOBLITZ. \( p \) -adic Numbers, \( p \) -adic Analysis, and Zeta-Functions. 2nd ed. 59 Lang. Cyclotomic Fields. 60 Arnold. Mathematical Methods in Classical Mechanics. 2nd ed. Douglas S. Bridges # Foundations of Real and Abstract Analysis Douglas S. Bridges Department of Mathematics University of Waikato Hamilton New Zealand Editorial Board S. Axler F.W. Gehring K.A. Ribet Mathematics Department Mathematics Department Department of Mathematics San Francisco State East Hall University of California University University of Michigan at Berkeley San Francisco, CA 94132 Ann Arbor, MI 48109 Berkeley, CA 94720 USA USA USA Mathematics Subject Classification (1991): 26-01, 28-01, 46Axx, 46-01 Library of Congress Cataloging-in-Publication Data Bridges, D.S. (Douglas S.), 1945- Foundations of real and abstract analysis / Douglas S. Bridges. p. cm. - (Graduate texts in mathematics ; 174) Includes index. ISBN 0-387-98239-6 (hardcover : alk. paper) 1. Mathematical analysis. I. Title. II. Series. QA300.B69 1997 \( {515} - \mathrm{{dc}}{21} \) 97-10649 (C) 1998 Springer-Verlag New York, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Dedicated to the memory of my parents: Douglas McDonald Bridges and Allison Hogg Sweet Analytics, 'tis thou hast ravished me. FAUSTUS (Marlowe) The stone which the builders refused is become the head stone of the corner. Psalm CXVIII, 22. ...from so simple a beginning endless forms most beautiful and most wonderful have been, and are being, evolved. THE ORIGIN OF SPECIES (Darwin) ## Preface The core of this book, Chapters 3 through 5, presents a course on metric, normed, and Hilbert spaces at the senior/graduate level. The motivation for each of these chapters is the generalisation of a particular attribute of the Euclidean space \( {\mathbf{R}}^{n} \) : in Chapter 3, that attribute is distance; in Chapter 4, length; and in Chapter 5, inner product. In addition to the standard topics that, arguably, should form part of the armoury of any graduate student in mathematics, physics, mathematical economics, theoretical statistics,..., this part of the book contains many results and exercises that are seldom found in texts on analysis at this level. Examples of the latter are Wong's Theorem (3.3.12) showing that the Lebesgue covering property is equivalent to the uniform continuity property, and Motzkin's result (5.2.2) that a nonempty closed subset of Euclidean space has the unique closest point property if and only if it is convex. The sad reality today is that, perceiving them as one of the harder parts of their mathematical studies, students contrive to avoid analysis courses at almost any cost, in particular that of their own educational and technical deprivation. Many universities have at times capitulated to the negative demand of students for analysis courses and have seriously watered down their expectations of students in that area. As a result, mathematics majors are graduating, sometimes with high honours, with little exposure to anything but a rudimentary course or two on real and complex analysis, often without even an introduction to the Lebesgue integral. For that reason, and also in order to provide a reference for material that is used in later chapters, I chose to begin this book with a long chapter providing a fast-paced course of real analysis, covering convergence of sequences and series, continuity, differentiability, and (Riemann and Riemann-Stieltjes) integration. The inclusion of that chapter means that the prerequisite for the book is reduced to the usual undergraduate sequence of courses on calculus. (One-variable calculus would suffice, in theory, but a lack of exposure to more advanced calculus courses would indicate a lack of the mathematical maturity that is the hidden prerequisite for most senior/graduate courses.) Chapter 2 is designed to show that the subject of differentiation does not end with the material taught in calculus courses, and to introduce the Lebesgue integral. Starting with the Vitali Covering Theorem, the chapter develops a theory of differentiation almost everywhere that underpins a beautiful approach to the Lebesgue integral due to F. Riesz [39]. One minor disadvantage of Riesz's approach is that, in order to handle multivariate integrals, it requires the theory of set-valued derivatives, a topic sufficiently involved and far from my intended route through elementary analysis that I chose to omit it altogether. The only place where this might be regarded as a serious omission is at the end of the chapter on Hilbert space, where I require classical vector integration to investigate the existence of weak solutions to the Dirichlet Problem in three-dimensional Euclidean space; since that investigation is only outlined, it seemed justifiable to rely only on the reader's presumed acquaintance with elementary vector calculus. Certainly, one-dimensional integration is all that is needed for a sound introduction to the \( {L}_{p} \) spaces of functional analysis, which appear in Chapter 4. Chapters 1 and 2 form Part I (Real Analysis) of the book; Part II (Abstract Analysis) comprises the remaining chapters and the appendices. I have already summarised the material covered in Chapters 3 through 5. Chapter 6, the final one, introduces functional analysis, starting with the Hahn-Banach Theorem and the consequent separation theorems. As well as the common elementary applications of the Hahn-Banach Theorem, I have included some deeper ones in duality theory. The chapter ends with the Baire Category Theorem, the Open Mapping Theorem, and their consequences. Here most of the applications are standard, although one or two unusual ones are included as exercises. The book has a preliminary section dealing with background material needed in the main text, and three appendices. The first appendix describes Bishop's construction of the real number line and the subsequent development of its basic algebraic and order properties; the

1008_(GTM174)Foundations of Real and Abstract Analysis

of functional analysis, which appear in Chapter 4. Chapters 1 and 2 form Part I (Real Analysis) of the book; Part II (Abstract Analysis) comprises the remaining chapters and the appendices. I have already summarised the material covered in Chapters 3 through 5. Chapter 6, the final one, introduces functional analysis, starting with the Hahn-Banach Theorem and the consequent separation theorems. As well as the common elementary applications of the Hahn-Banach Theorem, I have included some deeper ones in duality theory. The chapter ends with the Baire Category Theorem, the Open Mapping Theorem, and their consequences. Here most of the applications are standard, although one or two unusual ones are included as exercises. The book has a preliminary section dealing with background material needed in the main text, and three appendices. The first appendix describes Bishop's construction of the real number line and the subsequent development of its basic algebraic and order properties; the second deals briefly with axioms of choice and Zorn's Lemma; and the third shows how some of the material in the chapters—in particular, Minkowski’s Separation Theorem-can be used in the theory of Pareto optimality and competitive equilibria in mathematical economics. Part of my motivation in writing Appendix C was to indicate that "mathematical economics" is a far deeper subject than is suggested by the undergraduate texts on calculus and linear algebra that are published under that title. I have tried, wherever possible, to present proofs so that they translate mutatis mutandis into their counterparts in a more abstract setting, such as that of a metric space (for results in Chapter 1) or a topological space (for results in Chapter 3). On the other hand, some results first appear as exercises in one context before reappearing as theorems in another: one example of this is the Uniform Continuity Theorem, which first appears as \( {}^{1} \) Exercise (1.4.8: 8) in the context of a compact interval of \( \mathbf{R} \), and which is proved later, as Corollary (3.3.13), in the more general setting of a compact metric space. I hope that this procedure of double exposure will enable students to grasp the material more firmly. The text covers just over 300 pages, but the book is, in a sense, much larger, since it contains nearly 750 exercises, which can be classified into at least the following, not necessarily exclusive, types: - applications and extensions of the main propositions and theorems; - results that fill in gaps in proofs or that prepare for proofs later in the book; - pointers towards new branches of the subject; - deep and difficult challenges for the very best students. The instructor will have a wide choice of exercises to set the students as assignments or test questions. Whichever ones are set, as with the learning of any branch of mathematics it is essential that the student attempt as many exercises as the constraints of time, energy, and ability permit. It is important for the instructor/student to realise that many of the exercises-especially in Chapters 1 and 2-deal with results, sometimes major ones, that are needed later in the book. Such an exercise may not clearly identify itself when it first appears; if it is not attempted then, it will provide revision and reinforcement of that material when the student needs to tackle it later. It would have been unreasonable of me to have included major results as exercises without some guidelines for the solution of the nonroutine ones; in fact, a significant proportion of the exercises of all types come with some such guideline, even if only a hint. Although Chapters 3 through 6 make numerous references to Chapters 1 and 2, I have tried to make it easy for the reader to tackle the later chapters without ploughing through the first two. In this way the book can be used as a text for a semester course on metric, normed, and Hilbert spaces. (If \( {}^{1} \) A reference of the form Proposition (a.b.c) is to Proposition \( c \) in Section \( b \) of Chapter \( a \) ; one to Exercise \( \left( {a.b.c : d}\right) \) is to the \( d \) th exercise in the set of exercises with reference number (a.b.c); and one to (B3) is to the 3rd result in Appendix B. Within each section, displays that require reference indicators are numbered in sequence: \( \left( 1\right) ,\left( 2\right) ,\ldots \) . The counter for this numbering is reset at the start of a new section. Chapter 2 is not covered, the instructor may need to omit material that depends on familiarity with the Lebesgue integral-in particular Section 4 of Chapter 4.) Chapter 6 could be included to round off an introductory course on functional analysis. Chapter 1 could be used on its own as a second course on real analysis (following the typical advanced calculus course that introduces formal notions of convergence and continuity); it could also be used as a first course for senior students who have not previously encountered rigorous analysis. Chapters 1 and 2 together would make a good course on real variables, in preparation for either the material in Chapters 3 through 5 or a course on measure theory. The whole book could be used for a sequence of courses starting with real analysis and culminating in an introduction to functional analysis. I have drawn on the resource provided by many excellent existing texts cited in the bibliography, as well as some original papers (notably [39], in which Riesz introduced the development of the Lebesgue integral used in Chapter 2). My first drafts were prepared using the \( {T}^{3} \) Scientific Word Processing System; the final version was produced by converting the drafts to TEX and then using Scientific Word. Both \( {T}^{3} \) and Scientific Word are products of TCI Software Research, Inc. I am grateful to the following people who have helped me in the preparation of this book: - Patrick Er, who first suggested that I offer a course in analysis for economists, which mutated into the regular analysis course from which the book eventually emerged; - the students in my analysis classes from 1990 to 1996, who suffered various slowly improving drafts; — Cris Calude, Nick Dudley Ward, Mark Schroder, Alfred Seeger, Doru Stefanescu, and Wang Yuchuan, who read and commented on parts of the book; - the wonderfully patient and cooperative staff at Springer-Verlag; - my wife and children, for their patience (in more than one sense). It is right and proper for me here to acknowledge my unspoken debt of gratitude to my parents. This book really began 35 years ago, when, with their somewhat mystified support and encouragement, I was beginning my love affair with mathematics and in particular with analysis. It is sad that they did not live to see its completion. ## Contents Preface ix Introduction 1 I Real Analysis 9 1 Analysis on the Real Line 11 1.1 The Real Number Line 11 1.2 Sequences and Series 20 1.3 Open and Closed Subsets of the Line 35 1.4 Limits and Continuity 41 1.5 Calculus 53 2 Differentiation and the Lebesgue Integral 79 2.1 Outer Measure and Vitali's Covering Theorem 79 2.2 The Lebesgue Integral as an Antiderivative 93 2.3 Measurable Sets and Functions 110 123 3 Analysis in Metric Spaces 125 3.1 Metric and Topological Spaces 125 3.2 Continuity, Convergence, and Completeness 135 ## II Abstract Analysis xiv Contents 3.3 Compactness 146 3.4 Connectedness 158 3.5 Product Metric Spaces 165 4 Analysis in Normed Linear Spaces 173 4.1 Normed Linear Spaces 174 4.2 Linear Mappings and Hyperplanes 182 4.3 Finite-Dimensional Normed Spaces 189 4.4 The \( {L}_{p} \) Spaces 194 4.5 Function Spaces 204 4.6 The Theorems of Weierstrass and Stone 212 4.7 Fixed Points and Differential Equations 219 5 Hilbert Spaces 233 5.1 Inner Products 233 5.2 Orthogonality and Projections 237 5.3 The Dual of a Hilbert Space 252 6 An Introduction to Functional Analysis 259 6.1 The Hahn-Banach Theorem 259 6.2 Separation Theorems 275 6.3 Baire's Theorem and Beyond 279 A What Is a Real Number? 291 B Axioms of Choice and Zorn's Lemma 299 C Pareto Optimality 303 References 311 Index 317 ## Introduction What we now call analysis grew out of the calculus of Newton and Leibniz, was developed throughout the eighteenth century (notably by Euler), and slowly became logically sound (rigorous) through the work of Gauss, Cauchy, Riemann, Weierstrass, Lebesgue, and many others in the nineteenth and early twentieth centuries. Roughly, analysis may be characterised as the study of limiting processes within mathematics. These processes traditionally include the convergence of infinite sequences and series, continuity, differentiation, and integration, on the real number line \( \mathbf{R} \) ; but in the last 100 years analysis has moved far from the one- or finite-dimensional setting, to the extent that it now deals largely with limiting processes in infinite-dimensional spaces equipped with structures that produce meaningful abstractions of such notions as limit and continuous. Far from being merely the fantastical delight of mathematicians, these infinite-dimensional abstractions have served both to clarify phenomena whose true nature is often obscured by the peculiar structure of \( \mathbf{R} \), and to provide foundations for quantum physics, equilibrium economics, numerical approximation—indeed, a host of areas of pure and applied mathematics. So important is analysis that it is no exaggeration to describe as seriously deficient any honours graduate in physics, mathematics, or theoretical economics who has not had good exposure to at least the fundamentals of metric, normed, and Hilbert space theory, if not the next step, in which metric notions all but disappear in the further abstraction of topological spaces. Like many students of mathematics, even very good ones, you may find it hard to see the point of analysis, in which intuition often seems sacrificed to th

1008_(GTM174)Foundations of Real and Abstract Analysis

h structures that produce meaningful abstractions of such notions as limit and continuous. Far from being merely the fantastical delight of mathematicians, these infinite-dimensional abstractions have served both to clarify phenomena whose true nature is often obscured by the peculiar structure of \( \mathbf{R} \), and to provide foundations for quantum physics, equilibrium economics, numerical approximation—indeed, a host of areas of pure and applied mathematics. So important is analysis that it is no exaggeration to describe as seriously deficient any honours graduate in physics, mathematics, or theoretical economics who has not had good exposure to at least the fundamentals of metric, normed, and Hilbert space theory, if not the next step, in which metric notions all but disappear in the further abstraction of topological spaces. Like many students of mathematics, even very good ones, you may find it hard to see the point of analysis, in which intuition often seems sacrificed to the demon of rigour. Is our intuition-algebraic, arithmetic, and geometric-not a sufficiently good guide to mathematical reality in most cases? Alas, it is not, as is illustrated by considering the differentiability of functions. (We are assuming here that you are familiar with the derivative from elementary calculus courses.) When you first met the derivative, you probably thought that any continuous (real-valued) function—that is, loosely, one with an unbroken graph— on an interval of \( \mathbf{R} \) has a derivative at all points of its domain; in other words, its graph has a tangent everywhere. Once you came across simple examples, like the absolute value function \( x \mapsto \left| x\right| \), of functions whose graphs are unbroken but have no tangent at some point, it would have been natural to conjecture that if the graph were unbroken, then it had a tangent at all but a finite number of points. If you were really smart, you might even have produced an example of a continuous function, made up of lots of spikes, which was not differentiable at any of a sequence of points. This is about as far as intuition can go. But, as Weierstrass showed in the last century, and as you are invited to demonstrate in Exercise (1.5.1: 2), there exist continuous functions on \( \mathbf{R} \) whose derivative does not exist anywhere. Even this is not the end of the story: in a technical sense discussed in Chapter 6, most continuous functions on \( \mathbf{R} \) are nowhere differentiable! Here, then, is a dramatic failure of our intuition. We could give examples of many others, all of which highlight the need for the sort of careful analysis that is the subject of this book. Of course, analysis is not primarily concerned with pathological examples such as Weierstrass's one of a continuous, nowhere differentiable function. Its main aim is to build up a body of concepts, theorems, and proofs that describe a large part of the mathematical world (roughly, the continuous part) and are well suited to the mathematical demands of physicists, economists, statisticians, and others. The central chapters of this book, Chapters 3 through 5, give you an introduction to some of the fundamental concepts and results of modern analysis. The earlier chapters serve either as a background reference for the later ones or, if you have not studied much real analysis before, as a rapid introduction to that topic, in preparation for the rest of the book. The final chapter introduces some of the main themes of functional analysis, the study of continuous linear mappings on infinite-dimensional spaces. Having understood Chapters 3 through 6, you should be in a position to appreciate such other jewels of modern analysis as - approximation theory, in which complicated types of functions are approximated by more tractable ones such as polynomials of fixed maximum degree; - spectral theory of linear operators on a Hilbert space, generalising the theory of eigenvalues and eigenvectors of matrices; - analysis of one and several complex variables; - duality theory in topological vector spaces; - Haar measure and duality on locally compact groups, and the associated abstract generalisation of the Fourier transform; - \( {C}^{ * } \) - and von Neumann algebras of operators on a Hilbert space, providing rigorous foundations for quantum mechanics; - the theory of partial differential equations and the related potential problems of classical physics; - the calculus of variations and optimisation theory. These, however, are the subjects of other books. The time has come to begin this one by outlining the background material needed in the main chapters. Throughout this book, we assume familiarity with the fundamentals of informal set theory, as found in [20]. We use the following notation for sets of numbers. The set of natural numbers: \( \;\mathbf{N} = \{ 0,1,2,\ldots \} \) . The set of positive integers: \( {\mathbf{N}}^{ + } = \{ 1,2,3,\ldots \} \) . The set of integers: \( \;\mathbf{Z} = \{ 0, - 1,1, - 2,2,\ldots \} \) . The set of rational numbers: \( \;\mathbf{Q}\; = \;\left\{ {\pm \frac{m}{n} : m, n \in \mathbf{N}, n \neq 0}\right\} \) . For the purposes of this preliminary section only, we accept as given the algebraic and order properties of the set \( \mathbf{R} \) of real numbers, even though these are not introduced formally until Chapter 1. When the rule and domain describing a function \( f : A \rightarrow B \) are known or clearly understood, we may denote \( f \) by \[ x \mapsto f\left( x\right) . \] Note that we use the arrow \( \rightarrow \) as in "the function \( f : A \rightarrow B \) ", and the barred arrow \( \mapsto \) as in "the function \( x \mapsto {x}^{3} \) on \( \mathbf{R} \) ". We regard two functions with the same rule but different domains as different functions. In fact, we define two functions \( f \) and \( g \) to be equal if and only if 4 Introduction - they have the same domain and - \( f\left( x\right) = g\left( x\right) \) for each \( x \) in that domain. Thus the function \( x \mapsto {x}^{2} \) with domain \( \mathbf{N} \) is not the same as the function \( x \mapsto {x}^{2} \) with domain \( \mathbf{R} \) . When considering a rule that defines a function, we usually take the domain of the function as the set of all objects \( x \) (or at least all \( x \) of the type we wish to consider) to which the rule can be applied. For example, if we are working in the context of \( \mathbf{R} \), we consider the domain of the function \( x \mapsto 1/\left( {x - 1}\right) \) to be the set consisting of all real numbers other than 1. We sometimes give explicit definitions of functions by cases. For example, \[ f\left( x\right) = \left\{ \begin{array}{ll} 0 & \text{ if }x\text{ is rational } \\ 1 & \text{ if }x\text{ is irrational } \end{array}\right. \] defines a function \( f : \mathbf{R} \rightarrow \{ 0,1\} \) . A sequence is just a special kind of function: namely, one of the form \( n \mapsto {x}_{n} \) with domain \( {\mathbf{N}}^{ + };{x}_{n} \) is then called the \( n \) th term of the sequence. We denote by \( {\left( {x}_{n}\right) }_{n = 1}^{\infty } \), or \( \left( {{x}_{1},{x}_{2},\ldots }\right) \), or even just \( \left( {x}_{n}\right) \), the sequence whose \( n \) th term is \( {x}_{n} \) . (Of course, \( n \) is a dummy variable here; so, for example, \( \left. {\left( {x}_{k}\right) \text{is the same sequence as}\left( {x}_{n}\right) \text{.}}\right) \) If all the terms of \( \left( {x}_{n}\right) \) belong to a set \( X \), we refer to \( \left( {x}_{n}\right) \) as a sequence in \( X \) . We also apply the word "sequence", and notations such as \( {\left( {x}_{n}\right) }_{n = \nu }^{\infty } \), to a mapping \( n \mapsto {x}_{n} \) whose domain has the form \( \{ n \in \mathbf{Z} : n \geq \nu \} \) for some integer \( \nu \) . A subsequence of \( \left( {x}_{n}\right) \) is a sequence of the form \[ {\left( {x}_{{n}_{k}}\right) }_{k = 1}^{\infty } = \left( {{x}_{{n}_{1}},{x}_{{n}_{2}},{x}_{{n}_{3}},\ldots }\right) , \] where \( {n}_{1} < {n}_{2} < {n}_{3} < \cdots \) . More generally, if \( f \) is a one-one mapping of \( {\mathbf{N}}^{ + } \) into itself, we write \( {\left( {x}_{f\left( n\right) }\right) }_{n = 1}^{\infty } \), or even just \( \left( {x}_{f\left( n\right) }\right) \), to denote the sequence whose \( n \) th term is \( {x}_{f\left( n\right) } \) . This enables us, in Section 2 of Chapter 1, to make sense of an expression like \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{x}_{f\left( n\right) } \), denoting a rearrangement of the infinite series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{x}_{n} \) . By a finite sequence we mean an ordered \( n \) -tuple \( \left( {{x}_{1},\ldots ,{x}_{n}}\right) \), where \( n \) is any positive integer. A nonempty set \( X \) is said to be countable, or to have countably many elements, if it is the range of a sequence. Note that a nonempty finite set is countable according to this definition. An infinite countable set is said to be countably infinite. We regard the empty set as being both finite and countable. A set that is not countable is said to be uncountable, and to have uncountably many elements. Let \( f, g \) be mappings from subsets of a set \( X \) into a set \( Y \), where \( Y \) is equipped with a binary operation \( \diamond \) . We introduce the corresponding pointwise operation \( \diamond \) on \( f \) and \( g \) by setting \[ \left( {f\diamond g}\right) \left( x\right) = f\left( x\right) \diamond g\left( x\right) \] whenever \( f\left( x\right) \) and \( g\left( x\right) \) are both defined. Thus, taking \( Y = \mathbf{R} \), we see that the (pointwise) sum of \( f \) and \( g \) is given by \[ \left( {f + g}\right) \left( x\right) = f\left( x\right) + g\left( x\right) \] if \( f\left( x\right) \) and \( g\left( x\right) \) are both defined; and that the (pointwise) quotient of \( f \) and \( g \) is given by \[ \le

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le according to this definition. An infinite countable set is said to be countably infinite. We regard the empty set as being both finite and countable. A set that is not countable is said to be uncountable, and to have uncountably many elements. Let \( f, g \) be mappings from subsets of a set \( X \) into a set \( Y \), where \( Y \) is equipped with a binary operation \( \diamond \) . We introduce the corresponding pointwise operation \( \diamond \) on \( f \) and \( g \) by setting \[ \left( {f\diamond g}\right) \left( x\right) = f\left( x\right) \diamond g\left( x\right) \] whenever \( f\left( x\right) \) and \( g\left( x\right) \) are both defined. Thus, taking \( Y = \mathbf{R} \), we see that the (pointwise) sum of \( f \) and \( g \) is given by \[ \left( {f + g}\right) \left( x\right) = f\left( x\right) + g\left( x\right) \] if \( f\left( x\right) \) and \( g\left( x\right) \) are both defined; and that the (pointwise) quotient of \( f \) and \( g \) is given by \[ \left( {f/g}\right) \left( x\right) = f\left( x\right) /g\left( x\right) \] if \( f\left( x\right) \) and \( g\left( x\right) \) are defined and \( g\left( x\right) \neq 0 \) . If \( X = {\mathbf{N}}^{ + } \), so that \( f = \left( {x}_{n}\right) \) and \( g = \left( {y}_{n}\right) \) are sequences, then we also speak of termwise operations; for example, the termwise product of \( f \) and \( g \) is the sequence \( {\left( {x}_{n}{y}_{n}\right) }_{n = 1}^{\infty } \) . Pointwise operations extend in the obvious ways to finitely many functions. In the case of a sequence \( {\left( {f}_{n}\right) }_{n = 1}^{\infty } \) of functions with values in a normed space (see Chapter 4), once we have introduced the notion of a series in a normed space, we interpret \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{f}_{n} \) in the obvious way. By a family of elements of a set \( X \) we mean a mapping \( \lambda \mapsto {x}_{\lambda } \) of a set \( L \), called the index set for the family, into \( X \) . We also denote such a family by \( {\left( {x}_{\lambda }\right) }_{\lambda \in L} \) . A family with index set \( {\mathbf{N}}^{ + } \) is, of course, a sequence. By a subfamily of a family \( {\left( {x}_{\lambda }\right) }_{\lambda \in L} \) we mean a family \( {\left( {x}_{\lambda }\right) }_{\lambda \in J} \) where \( J \subset L \) . If \( {\left( {S}_{\lambda }\right) }_{\lambda \in L} \) is a family of sets, we write \[ \mathop{\bigcup }\limits_{{\lambda \in L}}{S}_{\lambda } = \left\{ {x : \exists \lambda \in L\left( {x \in {S}_{\lambda }}\right) }\right\} \] \[ \mathop{\bigcap }\limits_{{\lambda \in L}}{S}_{\lambda } = \left\{ {x : \forall \lambda \in L\left( {x \in {S}_{\lambda }}\right) }\right\} \] and we call \( \mathop{\bigcup }\limits_{{\lambda \in L}}{S}_{\lambda } \) and \( \mathop{\bigcap }\limits_{{\lambda \in L}}{S}_{\lambda } \), respectively, the union and the intersection of the family \( {\left( {S}_{\lambda }\right) }_{\lambda \in L} \) . We need some information about order relations on a set. (For fuller information about orders in general see Chapter 1 of [9].) A binary relation \( R \) on a set \( X \) is said to be - reflexive if \[ \forall a \in X\left( {aRa}\right) \] - irreflexive if \[ \forall a \in X\left( {\operatorname{not}\left( {aRa}\right) }\right) ; \] - symmetric if \[ \forall a, b \in X\left( {{aRb} \Rightarrow {bRa}}\right) ; \] Introduction - asymmetric if \[ \forall a, b \in X\left( {{aRb} \Rightarrow \operatorname{not}\left( {bRa}\right) }\right) \] - antisymmetric if \[ \forall a, b \in X\left( {\left( {{aRb}\text{ and }{bRa}}\right) \Rightarrow a = b}\right) ; \] - transitive if \[ \forall a, b, c \in X\left( {\left( {{aRa}\text{ and }{bRc}}\right) \Rightarrow {aRc}}\right) ; \] - total if \[ \forall a, b \in X\left( {{aRb}\text{ or }{bRa}}\right) . \] We use \( \succcurlyeq \) to represent a reflexive relation, and \( \succ \) to represent an irreflex- ive one. The notation \( a \preccurlyeq b \) (respectively, \( a \prec b \) ) is equivalent to \( b \succcurlyeq a \) (respectively, \( b \succ a \) ). When dealing with the usual order relations on the real line \( \mathbf{R} \), we use the standard symbols \( \geq , > , \leq , < \) instead of \( \succcurlyeq , \succ , \preccurlyeq , \prec \) , respectively. A binary relation \( R \) on a set \( X \) is said to be - a preorder if it is reflexive and transitive; - an equivalence relation if it is a symmetric preorder (in which case \( X \) is partitioned into disjoint equivalence classes, each equivalence class consisting of elements that are related under \( R \), and the set of these equivalence classes, written \( X/R \), is called the quotient set for \( R \) ); - a partial order if it is an antisymmetric preorder; - a total order if it is a total partial order; - a strict partial order if it is asymmetric and transitive - or, equivalently, if it is irreflexive and transitive. If \( R \) is a partial order on \( X \), we call the pair \( \left( {X, R}\right) \) -or, when there is no risk of confusion, just the set \( X \) itself - a partially ordered set. With each preorder \( \succcurlyeq \) on \( X \) we associate a strict partial order \( \succ \) and an equivalence relation \( \sim \) on \( X \), defined as follows. \[ x \succ y\;\text{ if and only if }\;x \succcurlyeq y\text{ and not }\left( {y \succcurlyeq x}\right) ; \] \[ x \sim y\;\text{ if and only if }\;x \succcurlyeq y\text{ and }y \succcurlyeq x. \] If \( \succcurlyeq \) is a total order, we have the Law of Trichotomy: \[ \forall x, y, z \in X\left( {x \succ y\text{ or }x = y\text{ or }x \prec y}\right) . \] Let \( S \) be a nonempty subset of a partially ordered set \( \left( {X, \succcurlyeq }\right) \) . An element \( B \in X \) is called an upper bound, or majorant, of \( S \) (relative to \( \succcurlyeq \) ) if \( B \succcurlyeq x \) for all \( x \in S \) . If there exist upper bounds of \( S \), then we say that \( S \) is bounded above, or majorised. An element \( B \in X \) is called a least upper bound, or supremum, of \( S \) if the following two conditions are satisfied. - \( B \) is an upper bound of \( S \) ; - if \( {B}^{\prime } \) is an upper bound of \( S \), then \( {B}^{\prime } \succcurlyeq B \) . Note that \( S \) has at most one supremum: for if \( B,{B}^{\prime } \) are suprema of \( S \), then \( {B}^{\prime } \succcurlyeq B \succcurlyeq {B}^{\prime } \) and so \( {B}^{\prime } = B \), by the antisymmetry of \( \succcurlyeq \) . If the supremum of \( S \) exists, we denote it by \( \sup S \) . We also denote it by \[ \mathop{\sup }\limits_{{1 \leq i \leq n}}{x}_{i},\max S,\mathop{\max }\limits_{{1 \leq i \leq n}}{x}_{i},\text{ or }{x}_{1} \vee {x}_{2} \vee \cdots \vee {x}_{n} \] if \( S = \left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} \) is a finite set, and by \[ \mathop{\sup }\limits_{{n \geq 1}}{x}_{n}\text{ or }\mathop{\bigvee }\limits_{{n = 1}}^{\infty }{x}_{n} \] if \( S = \left\{ {{x}_{1},{x}_{2},\ldots }\right\} \) is a countable set; we use similar notations without further comment. An upper bound of \( S \) that belongs to \( S \) is called a maximum element of \( S \), and is then a least upper bound of \( S \) . The maximum element, if it exists, of \( S \) is also called the largest, or greatest, element of \( S \) . An element \( b \in X \) is called a lower bound, or minorant, of \( S \) (relative to \( \succcurlyeq \) ) if \( x \succcurlyeq b \) for all \( x \in S \) . If there exist lower bounds of \( S \), then we say that \( S \) is bounded below, or minorised. An element \( b \in X \) is called a greatest lower bound, or infimum, of \( S \) if the following two conditions are satisfied. - \( b \) is a lower bound of \( S \) ; - if \( {b}^{\prime } \) is a lower bound of \( S \), then \( b \succcurlyeq {b}^{\prime } \) . \( S \) has at most one infimum, which we denote by \( \inf S \) . When describing infima, we also use such notations as \[ \mathop{\inf }\limits_{{1 \leq i \leq n}}{x}_{i},\min S,\mathop{\min }\limits_{{1 \leq i \leq n}}{x}_{i},\text{ or }{x}_{1} \land {x}_{2} \land \cdots \land {x}_{n} \] if \( S = \left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} \) is a finite set, and \[ \mathop{\inf }\limits_{{n \geq 1}}{x}_{n}\text{ or }\mathop{\bigwedge }\limits_{{n = 1}}^{\infty }{x}_{n} \] if \( S = \left\{ {{x}_{1},{x}_{2},\ldots }\right\} \) is a countable set. A lower bound of \( S \) that belongs to \( S \) is called a minimum element of \( S \), and is a greatest lower bound of \( S \) . The minimum element, if it exists, of \( S \) is also called the smallest, or least, element of \( S \) . The usual partial order \( \geq \) on \( \mathbf{R} \) gives rise to important operations on functions. If \( f, g \) are real-valued functions, we write \( f \geq g \) (or \( g \leq f \) ) to indicate that \( f\left( x\right) \geq g\left( x\right) \) for all \( x \) common to the domains of \( f \) and \( g \) . Regarding \( \vee \) and \( \land \) as binary operations on \( \mathbf{R} \), we define the corresponding functions \( f \vee g \) and \( f \land g \) as special cases of the notion \( f\diamond g \) previously introduced. By extension of these ideas, if \( {\left( {f}_{n}\right) }_{n = 1}^{\infty } \) is a sequence of real-valued functions, then the functions \( \mathop{\bigvee }\limits_{{n = 1}}^{\infty }{f}_{n} \) and \( \mathop{\bigwedge }\limits_{{n = 1}}^{\infty }{f}_{n} \) are defined by \[ \left( {\mathop{\bigvee }\limits_{{n = 1}}^{\infty }{f}_{n}}\right) \left( x\right) = \mathop{\bigvee }\limits_{{n = 1}}^{\infty }{f}_{n}\left( x\right) \] \[ \left( {\mathop{\bigwedge }\limits_{{n = 1}}^{\infty }{f}_{n}}\right) \left( x\right) = \mathop{\bigwedge }\limits_{{n = 1}}^{\infty }{f}_{n}\left( x\right) \] whenever the right-hand sides of these equations make sense. Now let \( f \) be a mapping of a set \( X \) into the partially ordered set \( \left( {\mathbf{R}, \geq }\rig

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. Regarding \( \vee \) and \( \land \) as binary operations on \( \mathbf{R} \), we define the corresponding functions \( f \vee g \) and \( f \land g \) as special cases of the notion \( f\diamond g \) previously introduced. By extension of these ideas, if \( {\left( {f}_{n}\right) }_{n = 1}^{\infty } \) is a sequence of real-valued functions, then the functions \( \mathop{\bigvee }\limits_{{n = 1}}^{\infty }{f}_{n} \) and \( \mathop{\bigwedge }\limits_{{n = 1}}^{\infty }{f}_{n} \) are defined by \[ \left( {\mathop{\bigvee }\limits_{{n = 1}}^{\infty }{f}_{n}}\right) \left( x\right) = \mathop{\bigvee }\limits_{{n = 1}}^{\infty }{f}_{n}\left( x\right) \] \[ \left( {\mathop{\bigwedge }\limits_{{n = 1}}^{\infty }{f}_{n}}\right) \left( x\right) = \mathop{\bigwedge }\limits_{{n = 1}}^{\infty }{f}_{n}\left( x\right) \] whenever the right-hand sides of these equations make sense. Now let \( f \) be a mapping of a set \( X \) into the partially ordered set \( \left( {\mathbf{R}, \geq }\right) \) . We say that \( f \) is bounded above on \( X \) if \[ f\left( X\right) = \{ f\left( x\right) : x \in X\} \] is bounded above as a subset of \( Y \) . We call \( \sup f\left( X\right) \), if it exists, the supremum of \( f \) on \( X \), and we denote it by \( \sup f,\mathop{\sup }\limits_{{x \in X}}f\left( x\right) \), or, in the case where \( X \) is a finite set, \( \max f \) . We also use obvious variations on these notations, such as \( \mathop{\sup }\limits_{{n \geq 1}}f\left( n\right) \) when \( X = {\mathbf{N}}^{ + } \) . We adopt analogous definitions and notations for bounded below on \( X \), infimum of \( f \), inf \( f \), and \( \min f \) . Finally, let \( f \) be a mapping of a partially ordered set \( \left( {X, \succcurlyeq }\right) \) into the partially ordered set \( \left( {\mathbf{R}, \geq }\right) \) . We say that \( f \) is - increasing if \( f\left( x\right) \geq f\left( {x}^{\prime }\right) \) whenever \( x \succcurlyeq {x}^{\prime } \) ; - strictly increasing if \( f\left( x\right) > f\left( {x}^{\prime }\right) \) whenever \( x \succ {x}^{\prime } \) ; - decreasing if \( f\left( x\right) \leq f\left( {x}^{\prime }\right) \) whenever \( x \succcurlyeq {x}^{\prime } \) ; and - strictly decreasing if \( f\left( x\right) < f\left( {x}^{\prime }\right) \) whenever \( x \succ {x}^{\prime } \) . Note that we use "increasing" and "strictly increasing" where some authors would use "nondecreasing" and "increasing", respectively. ## Part I ## Real Analysis 1 Analysis on the Real Line ...I will a round unvarnish'd tale deliver... OTHELLO, Act 1, Scene 3 In this chapter we provide a self-contained development of analysis on the real number line. We begin with an axiomatic presentation of \( \mathbf{R} \), from which we develop the elementary properties of exponential and logarithmic functions. We then discuss the convergence of sequences and series, paying particular attention to applications of the completeness of \( \mathbf{R} \) . Section 3 introduces open and closed sets, and lays the groundwork for later abstraction in the context of a metric space. Section 4 deals with limits and continuity of real-valued functions; the Heine-Borel-Lebesgue and Bolzano-Weierstrass theorems prepare us for the general, and extremely useful, notion of compactness, which is discussed in Chapter 3. The final section deals with the differential and integral calculus, a subject that is reviewed from a more advanced standpoint in Chapter 2. ## 1.1 The Real Number Line Although it is possible to construct the real number line \( \mathbf{R} \) from \( \mathbf{N} \) using elementary properties of sets and functions, in order to take us quickly to the heart of real analysis we relegate such a construction to Appendix A and instead present a set of axioms sufficient to characterise \( \mathbf{R} \) . These axioms fall into three categories: the first introduces the algebra of real numbers; the remaining two are concerned with the ordering on \( \mathbf{R} \) . Axiom R1. R is a field-that is, there exist a binary operation \( \left( {x, y}\right) \mapsto x + y \) of addition on \( \mathbf{R} \) , a binary operation \( \left( {x, y}\right) \mapsto {xy} \) of multiplication \( {}^{1} \) on \( \mathbf{R} \) , distinguished elements \( 0 \) (zero) and 1 (one) of \( \mathbf{R} \), with \( 0 \neq 1 \) , a unary operation \( x \mapsto - x \) (negation) on \( \mathbf{R} \), and a unary operation \( x \mapsto {x}^{-1} \) of reciprocation, or inversion, on \( \mathbf{R} \smallsetminus \{ 0\} \) such that for all \( x, y, z \in \mathbf{R} \) , \[ x + y = y + x \] \[ \left( {x + y}\right) + z = x + \left( {y + z}\right) , \] \[ 0 + x = x, \] \[ x + \left( {-x}\right) = 0 \] \[ {xy} = {yx} \] \[ \left( {xy}\right) z = x\left( {yz}\right) \] \[ x\left( {y + z}\right) = {xy} + {xz} \] \[ {1x} = x\text{, and} \] \[ x{x}^{-1} = 1\text{if}x \neq 0\text{.} \] Of course, we also denote \( {x}^{-1} \) by \( \frac{1}{x} \) or \( 1/x \) . Axioms R2. R is endowed with a total partial order \( \geq \) (greater than or equal to), and hence an associated strict partial order \( > \) (greater than), such that - if \( x \geq y \), then \( x + z \geq y + z \), and - if \( x \geq 0 \) and \( y \geq 0 \), then \( {xy} \geq 0 \) . Axiom R3. The least-upper-bound principle: if a nonempty subset \( S \) of \( \mathbf{R} \) is bounded above relative to the relation \( \geq \), then it has a (unique) least upper bound. The elements of \( \mathbf{R} \) are called real numbers. We say that a real number \( x \) is - positive if \( x > 0 \) , - negative if \( - x > 0 \), and \( {}^{1} \) For clarity, we sometimes write \( x \cdot y \) or \( x \times y \) for the product \( {xy} \) . - nonnegative if \( x \geq 0 \) . We denote the set of positive real numbers by \( {\mathbf{R}}^{ + } \), and the set of nonnegative real numbers by \( {\mathbf{R}}^{0 + } \) . Many of the fundamental arithmetic and order properties of \( \mathbf{R} \) are immediate consequences of results in the elementary theories of fields and partial orders, respectively. A number of these, illustrating the interplay between the algebra and the ordering on \( \mathbf{R} \), are given in the next set of exercises. \( {}^{2} \) ## (1.1.1) Exercises Prove each of the following statements, where \( x, y,{x}_{i},{y}_{i}\left( {1 \leq i \leq n}\right) \) are real numbers. .1 If \( {x}_{i} \geq {y}_{i} \) for each \( i \), then \( \mathop{\sum }\limits_{{i = 1}}^{n}{x}_{i} \geq \mathop{\sum }\limits_{{i = 1}}^{n}{y}_{i} \) . If also \( {x}_{k} > {y}_{k} \) for some \( k \), then \( \mathop{\sum }\limits_{{i = 1}}^{n}{x}_{i} > \mathop{\sum }\limits_{{i = 1}}^{n}{y}_{i} \) . .2 \( x \geq y \) if and only if \( x + z \geq y + z \) for all \( z \in \mathbf{R} \) ; this remains true with each instance of \( \geq \) replaced by one of \( > \) . .3 If \( {x}_{i} \geq 0 \) for each \( i \) and \( \mathop{\sum }\limits_{{i = 1}}^{n}{x}_{i} = 0 \), then \( {x}_{1} = {x}_{2} = \cdots = {x}_{n} = 0 \) . .4 The following are equivalent: \( x \geq y, x - y \geq 0, - y \geq - x,0 \geq y - x \) ; these equivalences also hold with \( \geq \) replaced everywhere by \( > \) . .5 If \( x \geq y \) and \( z \geq 0 \), then \( {xz} \geq {yz} \) . .6 If \( x > 0 \) and \( y > 0 \), then \( {xy} > 0 \) ; if \( x > 0 \) and \( 0 > y \), then \( 0 > {xy} \) ; if \( 0 > x \) and \( 0 > y \), then \( {xy} > 0 \) ; and these results hold with \( > \) replaced everywhere by \( \geq \) . .7 \( {x}^{2} \geq 0 \), and \( {x}^{2} = 0 \) if and only if \( x = 0 \) . .8 If \( x > 0 \), then \( {x}^{-1} > 0 \) ; and if \( x < 0 \), then \( {x}^{-1} < 0 \) . .9 \( x \geq y \) if and only if \( {xz} \geq {yz} \) for all \( z > 0 \) . .10 \( x > y > 0 \) if and only if \( {y}^{-1} > {x}^{-1} > 0 \) . .11 \( \max \{ x, y\} \geq 0 \) if and only if \( x \geq 0 \) or \( y \geq 0 \) ; \( \max \{ x, y\} > 0 \) if and only if \( x > 0 \) or \( y > 0 \) . .12 \( \min \{ x, y\} \geq 0 \) if and only if \( x \geq 0 \) and \( y \geq 0 \) ; \( \min \{ x, y\} > 0 \) if and only if \( x > 0 \) and \( y > 0 \) . --- \( {}^{2} \) If you are comfortable with the elementary field and order properties of \( \mathbf{R} \) , then you can safely omit Exercises (1.1.1) and (1.1.2). --- .13 The mapping \[ n \mapsto {n1} = \left\{ \begin{array}{ll} 0 & \text{ if }n = 0 \\ \underset{n\text{ terms }}{\underbrace{1 + 1 + \cdots + 1}} & \text{ if }n \geq 1 \end{array}\right. \] from \( \mathbf{N} \) into \( \mathbf{R} \) is one-one and preserves order, addition, and multiplication. We use this mapping to identify \( \mathbf{N} \) with the subset \( \{ {n1} : n \in \mathbf{N}\} \) of R. In turn, we then identify a negative integer \( n \) with \( - \left( {-n}\right) 1 \), and a rational number \( m/n \) with the real number \( m{n}^{-1} \) . We make these identifications without further comment. .14 If \( S \) is a nonempty majorised set of integers, then \( m = \sup S \) is an integer. (Assume the contrary and obtain integers \( n,{n}^{\prime } \) such that \( m - 1 < n < {n}^{\prime } < m \) .) .15 There exists \( n \in \mathbf{Z} \) such that \( n - 1 \leq x < n \) . (If \( x \geq 0 \), apply the least-upper-bound principle to \( S = \{ k \in \mathbf{Z} : k \leq x\} \) .) .16 If \( x > 0 \) and \( y \geq 0 \), then there exists \( n \in {\mathbf{N}}^{ + } \) such that \( {nx} > y \) . (Consider \( \{ k \in \mathbf{N} : {kx} \leq y\} \) .) This important property is sometimes introduced as an axiom, the Axiom of Archimedes. .17 \( x > 0 \) if and only if there exists a positive integer \( n > {x}^{-1} \) . .18 \( x \geq 0 \) if and only if \( x \geq - 1/n \) for all positive integers \( n \) . .19 \( \mathbf{Q} \) is order dense in \( \mathbf{R} \) —that is, if \( x < y \), then there exists \( q \in \mathbf{Q} \) such that \( x < q < y \) . (Reduce to the case \( y > 0 \) . Choose in turn integers \( n

1008_(GTM174)Foundations of Real and Abstract Analysis

sed set of integers, then \( m = \sup S \) is an integer. (Assume the contrary and obtain integers \( n,{n}^{\prime } \) such that \( m - 1 < n < {n}^{\prime } < m \) .) .15 There exists \( n \in \mathbf{Z} \) such that \( n - 1 \leq x < n \) . (If \( x \geq 0 \), apply the least-upper-bound principle to \( S = \{ k \in \mathbf{Z} : k \leq x\} \) .) .16 If \( x > 0 \) and \( y \geq 0 \), then there exists \( n \in {\mathbf{N}}^{ + } \) such that \( {nx} > y \) . (Consider \( \{ k \in \mathbf{N} : {kx} \leq y\} \) .) This important property is sometimes introduced as an axiom, the Axiom of Archimedes. .17 \( x > 0 \) if and only if there exists a positive integer \( n > {x}^{-1} \) . .18 \( x \geq 0 \) if and only if \( x \geq - 1/n \) for all positive integers \( n \) . .19 \( \mathbf{Q} \) is order dense in \( \mathbf{R} \) —that is, if \( x < y \), then there exists \( q \in \mathbf{Q} \) such that \( x < q < y \) . (Reduce to the case \( y > 0 \) . Choose in turn integers \( n > 1/\left( {y - x}\right) \) and \( k \geq {ny} \), and let \( m \) be the least integer such that \( y \leq m/n \) . Show that \( x < \left( {m - 1}\right) /n < y \) .) .20 If \( S \) and \( T \) are nonempty majorised sets of positive numbers, then \[ \sup \{ {st} : s \in S, t \in T\} = \sup S \times \sup T. \] .21 The following are equivalent conditions on nonempty subsets \( X \) and \( Y \) of \( \mathbf{R} \) . (i) \( x \leq y \) for all \( x \in X \) and \( y \in Y \) . (ii) There exists \( \tau \in \mathbf{R} \) such that \( x \leq \tau \leq y \) for all \( x \in X \) and \( y \in Y \) . Each real number \( x \) has a corresponding absolute value, defined as \[ \left| x\right| = \max \{ x, - x\} . \] ## (1.1.2) Exercises Prove each of the following statements about real numbers \( x, y,\varepsilon \) . .1 \( \left| x\right| \geq 0 \), and \( \left| x\right| = 0 \) if and only if \( x = 0 \) . .2 \( \left| x\right| \leq \varepsilon \) if and only if \( - \varepsilon \leq x \leq \varepsilon \) . .3 \( \left| x\right| < \varepsilon \) if and only if \( - \varepsilon < x < \varepsilon \) . .4 \( x = 0 \) if and only if either \( \left| x\right| \leq \varepsilon \) for each \( \varepsilon > 0 \) or else \( \left| x\right| < \varepsilon \) for each \( \varepsilon > 0 \) . .5 \( \left| {x + y}\right| \leq \left| x\right| + \left| y\right| \) (triangle inequality). .6 \( \left| {x - y}\right| \geq \left| \right| x\left| -\right| y\left| \right| \) . .7 \( \left| {xy}\right| = \left| x\right| \left| y\right| \) . So far we have not indicated how useful the least-upper-bound principle is. In fact, it is not only useful, but essential: the field \( \mathbf{Q} \) of rational numbers, with its usual ordering \( > \), satisfies all the properties listed in axioms \( \mathbf{{R1}} \) and \( \mathbf{{R2}} \), so we need something more to distinguish \( \mathbf{R} \) from \( \mathbf{Q} \) . Moreover, without the least-upper-bound principle or some property equivalent to it, we cannot even prove that a positive real number has a square root. We now sketch how the least-upper-bound principle enables us to define \( {a}^{r} \) for any \( a > 0 \) and any \( r \in \mathbf{R} \) . When \( n \) is an integer, \( {a}^{n} \) is defined as in elementary algebra. So our first real task is to define \( {a}^{m/n} \) when \( m \) and \( n \) are nonzero integers; this we do by setting \[ {a}^{m/n} = \sup \left\{ {x \in \mathbf{R} : {x}^{n} < {a}^{m}}\right\} . \] (1) Of course, we are using the least-upper-bound principle here, so we must ensure that the set on the right-hand side of (1) is both nonempty and bounded above. To prove that it is nonempty, we use the Axiom of Archime- des (Exercise (1.1.1:16)) to find a positive integer \( k \) such that \( k{a}^{m} > 1 \) ; then \( {k}^{n}{a}^{m} \geq k{a}^{m} > 1 \), so \( {\left( 1/k\right) }^{n} < {a}^{m} \) . On the other hand, as \[ {\left( 1 + {a}^{m}\right) }^{n} \geq 1 + n{a}^{m} > {a}^{m}\;\text{if}n \geq 1\text{, and} \] \[ {\left( \frac{1}{1 + {a}^{m}}\right) }^{n} \geq 1 - n{a}^{m} > {a}^{m}\;\text{if }n \leq - 1, \] the set in question is bounded above (by \( 1 + {a}^{m} \) in the first case, and by \( 1/\left( {1 + {a}^{m}}\right) \) in the second). Hence \( {a}^{m/n} \) exists. Our first result enables us to prove some basic properties of \( {a}^{m/n} \) . (1.1.3) Lemma. Let \( a > 0 \) and \( s \) be real numbers, and \( m, n \) positive integers such that \( {s}^{n} < {a}^{m} \) . Then there exists \( t \in \mathbf{R} \) such that \( s < t \) and \( {t}^{n} < {a}^{m} \) . Proof. Using Exercise (1.1.1:16), choose a positive integer \( N \) such that \[ 0 < {N}^{-1} < \min \left\{ {1,{2}^{-n}{\left( 1 + \left| s\right| \right) }^{-n}\left( {{a}^{m} - {s}^{n}}\right) }\right\} . \] Writing \( t = s + {N}^{-1} \) and using the binomial theorem, we have \[ {t}^{n} = \mathop{\sum }\limits_{{k = 0}}^{n}\left( \begin{array}{l} n \\ k \end{array}\right) {s}^{n - k}{N}^{-k} \] \[ \leq {s}^{n} + \mathop{\sum }\limits_{{k = 1}}^{n}\left( \begin{array}{l} n \\ k \end{array}\right) {\left| s\right| }^{n - k}{N}^{-1} \] \[ < {s}^{n} + {N}^{-1}\mathop{\sum }\limits_{{k = 0}}^{n}\left( \begin{array}{l} n \\ k \end{array}\right) {\left( 1 + \left| s\right| \right) }^{n - k} \] \[ < {s}^{n} + {\left( 1 + \left| s\right| \right) }^{n}{N}^{-1}\mathop{\sum }\limits_{{k = 0}}^{n}\left( \begin{array}{l} n \\ k \end{array}\right) \] \[ = {s}^{n} + {2}^{n}{\left( 1 + \left| s\right| \right) }^{n}{N}^{-1} \] \[ < {a}^{m}, \] as we required. Taking \( s = 0 \) in this lemma, we see that \( {a}^{m/n} > 0 \) . The lemma also enables us to prove that \[ {\left( {a}^{m/n}\right) }^{n} = {a}^{m} \] (2) For if \( {\left( {a}^{m/n}\right) }^{n} < {a}^{m} \), then, by Lemma (1.1.3), there exists \( t > {a}^{m/n} \) such that \( {t}^{n} < {a}^{m} \), which contradicts the definition of \( {a}^{m/n} \) ; on the other hand, that same definition ensures that \( {\left( {a}^{m/n}\right) }^{n} \leq {a}^{m} \) and hence that (2) holds. Using (2) and methods familiar from elementary algebra courses, we can now prove the usual laws of indices, \[ {a}^{r}{a}^{s} = {a}^{r + s} \] \[ {\left( {a}^{r}\right) }^{s} = {a}^{rs} \] when the indices \( r, s \) are rational. We next extend the definition of \( {a}^{r} \) to cover all \( r \in \mathbf{R} \) . To begin with, we consider the case \( a > 1 \), when we define \[ {a}^{r} = \sup \left\{ {{a}^{q} : q \in \mathbf{Q}, q < r}\right\} . \] (3) It is left as an exercise to show that the set on the right-hand side of (3) is nonempty and bounded above, and that if \( r \) is rational, this definition gives \( {a}^{r} \) the same value as the one given by our earlier definition. We can now prove the laws of indices for arbitrary \( r, s \in \mathbf{R} \) . Taking the first law as an illustration, we observe that if \( u, v \) are rational numbers with \( u < r \) and \( v < s \), then \( u + v < r + s \), so \[ {a}^{u}{a}^{v} = {a}^{u + v} \leq {a}^{r + s}. \] By Exercise (1.1.1: 20), \[ {a}^{r}{a}^{s} = \sup \left\{ {{a}^{u} : u \in \mathbf{Q}, u < r}\right\} \times \sup \left\{ {{a}^{v} : v \in \mathbf{Q}, v < s}\right\} \] \[ = \sup \left\{ {{a}^{u}{a}^{v} : u, v \in \mathbf{Q}, u < r, v < s}\right\} \] \[ \leq {a}^{r + s}\text{.} \] On the other hand, if \( q \in \mathbf{Q} \) and \( q < r + s \), then we choose rational numbers \( u, v \) with \( u < r, v < s \), and \( q = u + v \) : to do so, we use Exercise (1.1.1: 19) to find \( u \in \mathbf{Q} \) with \( q - s < u < r \) and we then set \( v = q - u \) . We have \[ {a}^{q} = {a}^{u + v} = {a}^{u}{a}^{v} \leq {a}^{r}{a}^{s}. \] Hence \[ {a}^{r + s} = \sup \left\{ {{a}^{q} : q \in \mathbf{Q}, q < r + s}\right\} \leq {a}^{r}{a}^{s}, \] and therefore \( {a}^{r}{a}^{s} = {a}^{r + s} \) . It remains to define \[ {a}^{r} = \left\{ \begin{array}{ll} {\left( {a}^{-1}\right) }^{-r} & \text{if }0 < a < 1 \\ 1 & \text{if }a = 1 \end{array}\right. \] and to verify - routinely - that the laws of indices hold in these cases also. ## (1.1.4) Exercises .1 Let \( a > 1 \) and let \( r \in \mathbf{R} \) . Prove that \( \left\{ {{a}^{q} : q \in \mathbf{Q}, q < r}\right\} \) is nonempty and bounded above. Prove also that if \( r = m/n \) for integers \( m, n \) with \( n \neq 0 \), then definitions (1) and (3) give the same value for \( {a}^{r} \) . .2 Prove that if \( 0 < a \neq 1 \) and \( {a}^{x} = 1 \), then \( x = 0 \) . (Consider first the case where \( a > 1 \), and note that if \( q \in \mathbf{Q} \) and \( {a}^{q} \leq 1 \), then \( q \leq 0 \) .) .3 Let \( a > 0 \) and \( x > y \) . Prove that if \( a > 1 \), then \( {a}^{x} > {a}^{y} \) ; and that if \( a < 1 \), then \( {a}^{x} < {a}^{y} \) . .4 Prove that if \( a > 0 \), then for each \( x > 0 \) there exists a unique \( y \in \mathbf{R} \) such that \( {a}^{y} = x \) . (First take \( a > 1 \) and \( x > 1 \) . Write \( a = 1 + t \) and, by expanding \( {\left( 1 + t\right) }^{n} \), compute \( n \in {\mathbf{N}}^{ + } \) such that \( {a}^{n} > x \) . Then consider \( \left. {\left\{ {q \in \mathbf{Q} : {a}^{q} \leq x}\right\} \text{.}}\right) \) 1. Analysis on the Real Line .5 Let \( f \) be a strictly increasing mapping of \( \mathbf{R} \) onto \( {\mathbf{R}}^{ + } \) such that \( f\left( 0\right) = \) 1 and \( f\left( {x + y}\right) = f\left( x\right) f\left( y\right) \) . Prove that \( f\left( x\right) = {a}^{x} \), where \( a = f\left( 1\right) > 1 \) . (First prove that \( f\left( q\right) = {a}^{q} \) for all rational \( q \) .) If \( a > 0 \), Exercise (1.1.4:4) allows us to define \( {\log }_{a} \), the logarithmic function with base \( a \), as follows. For each \( x > 0 \) , \[ y = {\log }_{a}x\;\text{ if and only if }\;{a}^{y} = x. \] This function has domain \( {\mathbf{R}}^{ + } \) and maps \( {\mathbf{R}}^{ + } \) onto \( \mathbf{R} \) . From the laws of indices we easily deduce the laws of logarith

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y expanding \( {\left( 1 + t\right) }^{n} \), compute \( n \in {\mathbf{N}}^{ + } \) such that \( {a}^{n} > x \) . Then consider \( \left. {\left\{ {q \in \mathbf{Q} : {a}^{q} \leq x}\right\} \text{.}}\right) \) 1. Analysis on the Real Line .5 Let \( f \) be a strictly increasing mapping of \( \mathbf{R} \) onto \( {\mathbf{R}}^{ + } \) such that \( f\left( 0\right) = \) 1 and \( f\left( {x + y}\right) = f\left( x\right) f\left( y\right) \) . Prove that \( f\left( x\right) = {a}^{x} \), where \( a = f\left( 1\right) > 1 \) . (First prove that \( f\left( q\right) = {a}^{q} \) for all rational \( q \) .) If \( a > 0 \), Exercise (1.1.4:4) allows us to define \( {\log }_{a} \), the logarithmic function with base \( a \), as follows. For each \( x > 0 \) , \[ y = {\log }_{a}x\;\text{ if and only if }\;{a}^{y} = x. \] This function has domain \( {\mathbf{R}}^{ + } \) and maps \( {\mathbf{R}}^{ + } \) onto \( \mathbf{R} \) . From the laws of indices we easily deduce the laws of logarithms: \[ {\log }_{a}{xy} = {\log }_{a}x + {\log }_{a}y \] \[ {\log }_{a}\left( {x}^{r}\right) = r{\log }_{a}x \] \[ {\log }_{b}x = {\log }_{b}a \times {\log }_{a}x\text{, where}b > 0\text{.} \] Anticipating the theory of convergence of series from the next section, we introduce the number \[ \mathrm{e} = \mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{1}{n!} \] and call \( {\log }_{\mathrm{e}} \) the natural logarithmic function on \( {\mathbf{R}}^{ + } \) . It is customary to denote \( {\log }_{\mathrm{e}} \) by either \( \log \) or \( \ln \) . ## (1.1.5) Exercises .1 Prove the laws of logarithms. .2 Prove that if \( a > 1 \), then the function \( {\log }_{a} \) is strictly increasing; and that if \( 0 < a < 1 \), then \( {\log }_{a} \) is strictly decreasing. .3 Let \( a > 1 \), and let \( f \) be an increasing mapping of \( {\mathbf{R}}^{ + } \) into \( \mathbf{R} \) such that \( f\left( a\right) = 1 \) and \( f\left( {xy}\right) = f\left( x\right) + f\left( y\right) \) . Prove that \( f\left( x\right) = {\log }_{a}x \) . For convenience, we collect here the definitions of the various types of interval in \( \mathbf{R} \) . The open intervals are the sets of the following forms, where \( a, b \) are real numbers with \( a < b \) : \[ \left( {a, b}\right) = \{ x \in \mathbf{R} : a < x < b\} \] \[ \left( {a,\infty }\right) = \{ x \in \mathbf{R} : a < x\} \] \[ \left( {-\infty, b}\right) = \{ x \in \mathbf{R} : x < b\} \] \[ \left( {-\infty ,\infty }\right) = \mathbf{R}\text{.} \] The closed intervals are the sets of the following forms, where \( a, b \) are real numbers with \( a \leq b \) : \[ \left\lbrack {a, b}\right\rbrack = \{ x \in \mathbf{R} : a \leq x \leq b\} \] \[ \lbrack a,\infty ) = \{ x \in \mathbf{R} : a \leq x\} \] \[ ( - \infty, b\rbrack = \{ x \in \mathbf{R} : x \leq b\} \text{.} \] By convention, \( \mathbf{R} \) is regarded as both an open interval and a closed interval. The remaining types of interval are: \[ \text{half open on the left:}\;(a, b\rbrack = \{ x \in \mathbf{R} : a < x \leq b\} \text{,} \] \[ \text{half open on the right:}\lbrack a, b) = \{ x \in \mathbf{R} : a \leq x < b\} \text{.} \] Intervals of the form \( \left\lbrack {a, b}\right\rbrack ,\left( {a, b}\right) ,\lbrack a, b) \), or \( (a, b\rbrack \), where \( a, b \in \mathbf{R} \), are said to be finite or bounded, and to have left endpoint \( a \), right endpoint \( b \), and length \( b - a \) . Intervals of the remaining types are called infinite and are said to have length \( \infty \) . The length of any interval \( I \) is denoted by \( \left| I\right| \) . A bounded closed interval in \( \mathbf{R} \) is also called a compact interval. Finally, we define the complex numbers to be the elements of the set \( \mathbf{C} = \mathbf{R} \times \mathbf{R} \), with the usual equality and with algebraic operations of addition and multiplication defined, respectively, by the equations \[ \left( {x, y}\right) + \left( {{x}^{\prime },{y}^{\prime }}\right) = \left( {x + {x}^{\prime }, y + {y}^{\prime }}\right) , \] \[ \left( {x, y}\right) \times \left( {{x}^{\prime },{y}^{\prime }}\right) = \left( {x{x}^{\prime } - y{y}^{\prime }, x{y}^{\prime } + {x}^{\prime }y}\right) . \] Then \( x \mapsto \left( {x,0}\right) \) is a one-one mapping of \( \mathbf{R} \) onto the set \( \mathbf{C} \times \{ 0\} \) and is used to identify \( \mathbf{R} \) with that subset of \( \mathbf{C} \) . With this identification, we have \( {\mathrm{i}}^{2} = - 1 \), where \( \mathrm{i} \) is the complex number \( \left( {0,1}\right) \) ; so the complex number \( \left( {x, y}\right) \) can be identified with the expression \( x + \mathrm{i}y \) . The real numbers \( x \) and \( y \) are then called the real and imaginary parts of \( z = \left( {x, y}\right) \), respectively, and we write \[ x = \operatorname{Re}\left( {x, y}\right) \] \[ y = \operatorname{Im}\left( {x, y}\right) \text{.} \] The conjugate of \( z \) is \[ {z}^{ * } = \left( {x, - y}\right) = x - \mathrm{i}y \] and the modulus of \( z \) is \[ \left| z\right| = \sqrt{{x}^{2} + {y}^{2}} \] In the remainder of this book we assume the basic properties of the real and complex numbers such as those found in the foregoing exercises. 20 1. Analysis on the Real Line ## 1.2 Sequences and Series Although often relegated to a minor role in courses on real analysis, the theory of convergence of sequences and series in \( \mathbf{R} \) provides both a model for more abstract convergence theories such as those in our later chapters, and many important examples. It is convenient to introduce here two useful expressions about properties of positive integers. Let \( P\left( {m, n}\right) \) be a property applicable to pairs \( \left( {m, n}\right) \) of positive integers. If there exists \( N \) such that \( P\left( {m, n}\right) \) holds for all \( m, n \geq N \) , then we say that \( P\left( {m, n}\right) \) holds for all sufficiently large \( m \) and \( n \) . We interpret similarly the statement \( P\left( n\right) \) holds for all sufficiently large \( n \) , where \( P\left( n\right) \) is a property applicable to positive integers \( n \) . On the other hand, if for each positive integer \( i \) there exists a positive integer \( j > i \) such that \( P\left( j\right) \) holds, then we say that \( P\left( n\right) \) holds for infinitely many values of \( n \) . We say that a sequence \( {}^{3}\left( {a}_{n}\right) \) of real numbers converges to a real number \( a \), called the limit of \( \left( {a}_{n}\right) \), if for each \( \varepsilon > 0 \) there exists a positive integer \( N \), depending on \( \varepsilon \), such that \( \left| {a - {a}_{n}}\right| \leq \varepsilon \) whenever \( n \geq N \) . Thus \( \left( {a}_{n}\right) \) converges to \( a \) if and only if for each \( \varepsilon > 0 \) we have \( \left| {a - {a}_{n}}\right| \leq \varepsilon \) for all sufficiently large \( n \) . In that case we write \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}{a}_{n} = a \] or \[ {a}_{n} \rightarrow a\text{ as }n \rightarrow \infty , \] and we also say that \( {a}_{n} \) tends to \( a \) as \( n \rightarrow \infty \) . On the other hand, we say that \( \left( {a}_{n}\right) \) diverges to \( \infty \), and we write \[ {a}_{n} \rightarrow \infty \text{as}n \rightarrow \infty \text{,} \] if for each \( K > 0 \) we have \( {a}_{n} > K \) for all sufficiently large \( n \) . If for each \( K > 0 \) we have \( {a}_{n} < - K \) for all sufficiently large \( n \), then we say that \( \left( {a}_{n}\right) \) diverges to \( - \infty \), and we write \[ {a}_{n} \rightarrow - \infty \text{as}n \rightarrow \infty \text{.} \] ## (1.2.1) Exercises .1 Prove that if \( \left( {a}_{n}\right) \) converges to both \( a \) and \( {a}^{\prime } \), then \( a = {a}^{\prime } \) . (Show that \( \left| {a - {a}^{\prime }}\right| < \varepsilon \) for each \( \varepsilon > 0 \) . This exercise justifies the use of the definite article in the phrase "the limit of \( \left( {a}_{n}\right) \) ".) --- \( {}^{3} \) We can extend the definitions of convergence and divergence of sequences in the obvious ways to cover families of the form \( {\left( {a}_{n}\right) }_{n \geq \nu } \), where \( \nu \in \mathbf{Z} \) ; all that matters is that \( {a}_{n} \) be defined for all sufficiently large positive integers \( n \) . This observation makes sense of the last part of Proposition (1.2.2), where we discuss the limit of a quotient of two sequences. --- .2 Let \( c > 0 \) . Prove that \( \left( {a}_{n}\right) \) converges to \( a \) if and only if for each \( \varepsilon > 0 \) there exists a positive integer \( N \), depending on \( \varepsilon \), such that \( \left| {a - {a}_{n}}\right| \leq {c\varepsilon } \) for all \( n \geq N \) . .3 Prove that if a sequence \( \left( {a}_{n}\right) \) converges to a limit, then it is bounded, in the sense that there exists \( c > 0 \) such that \( \left| {a}_{n}\right| \leq c \) for all \( n \) . .4 Let \( r \in \mathbf{R} \), and let \( \left( {a}_{n}\right) \) be a convergent sequence in \( \mathbf{R} \) such that \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{a}_{n} > r \) . Prove that \( {a}_{n} > r \) for all sufficiently large \( n \) . .5 Let \( r \in \mathbf{R} \), and let \( \left( {a}_{n}\right) \) be a convergent sequence in \( \mathbf{R} \) such that \( {a}_{n} \geq r \) for all sufficiently large \( n \) . Prove that \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{a}_{n} \geq r \) . .6 Prove that if \( \left( {a}_{n}\right) \) diverges to infinity and \( \left( {b}_{n}\right) \) converges to a limit \( b \in \mathbf{R} \), then the sequence \( \left( {{a}_{n} + {b}_{n}}\right) \) diverges to infinity. The process of taking limits of sequences preserves the basic operations of arithmetic. (1.2.2) Proposition. Let \( \left( {a}_{n}\right) \) and \( \left( {b}_{n}\right) \) be sequences of real numbers converging to limits \(

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t| \leq c \) for all \( n \) . .4 Let \( r \in \mathbf{R} \), and let \( \left( {a}_{n}\right) \) be a convergent sequence in \( \mathbf{R} \) such that \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{a}_{n} > r \) . Prove that \( {a}_{n} > r \) for all sufficiently large \( n \) . .5 Let \( r \in \mathbf{R} \), and let \( \left( {a}_{n}\right) \) be a convergent sequence in \( \mathbf{R} \) such that \( {a}_{n} \geq r \) for all sufficiently large \( n \) . Prove that \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{a}_{n} \geq r \) . .6 Prove that if \( \left( {a}_{n}\right) \) diverges to infinity and \( \left( {b}_{n}\right) \) converges to a limit \( b \in \mathbf{R} \), then the sequence \( \left( {{a}_{n} + {b}_{n}}\right) \) diverges to infinity. The process of taking limits of sequences preserves the basic operations of arithmetic. (1.2.2) Proposition. Let \( \left( {a}_{n}\right) \) and \( \left( {b}_{n}\right) \) be sequences of real numbers converging to limits \( a \) and \( b \), respectively. Then as \( n \rightarrow \infty \) , \[ {a}_{n} + {b}_{n} \rightarrow a + b \] \[ {a}_{n} - {b}_{n} \rightarrow a - b \] \[ {a}_{n}{b}_{n} \rightarrow {ab} \] \[ \max \left\{ {{a}_{n},{b}_{n}}\right\} \rightarrow \max \{ a, b\} \] \[ \min \left\{ {{a}_{n},{b}_{n}}\right\} \rightarrow \min \{ a, b\} ,\text{ and } \] \[ \left| {a}_{n}\right| \rightarrow \left| a\right| \] If also \( b \neq 0 \), then \( {b}_{n} \neq 0 \) for all sufficiently large \( n \), and \( {a}_{n}/{b}_{n} \rightarrow a/b \) as \( n \rightarrow \infty \) . Proof. We prove only the last statement, leaving the other cases to Exercise (1.2.3: 1). Assume that \( b \neq 0 \) . Then, by Exercise (1.2.1: 4), there exists \( {N}_{0} \) such that \( \left| {b}_{n}\right| > \frac{1}{2}\left| b\right| \), and therefore \( {a}_{n}/{b}_{n} \) is defined, for all \( n \geq {N}_{0} \) . Given \( \varepsilon > 0 \) , choose \( N \geq {N}_{0} \) such that \( \left| {{a}_{n} - a}\right| < \varepsilon \) and \( \left| {{b}_{n} - b}\right| < \varepsilon \) for all \( n \geq N \) . For all such \( n \) we have \[ \left| {\frac{{a}_{n}}{{b}_{n}} - \frac{a}{b}}\right| = \frac{\left| b{a}_{n} - a{b}_{n}\right| }{\left| {b}_{n}\right| \left| b\right| } \] \[ \leq \frac{\left| b\left( {a}_{n} - a\right) + a\left( b - {b}_{n}\right) \right| }{\frac{1}{2}{\left| b\right| }^{2}} \] \[ \leq 2{\left| b\right| }^{-2}\left( {\left| b\right| \left| {{a}_{n} - a}\right| + \left| a\right| \left| {b - {b}_{n}}\right| }\right) \] \[ \leq 2{\left| b\right| }^{-2}\left( {\left| a\right| + \left| b\right| }\right) \varepsilon \] The result now follows from Exercise (1.2.1: 2). ## (1.2.3) Exercises .1 Prove the remaining parts of Proposition (1.2.2). .2 Prove that if \( k \geq 2 \) and \( \nu \geq 1 \) are integers, then \[ {\left( 1 + \frac{1}{k}\right) }^{k - 1} > \frac{3}{2}\text{ and }{\left( 1 + \frac{1}{k}\right) }^{\nu \left( {k - 1}\right) } > \nu . \] Hence prove that if \( 0 \leq \left| r\right| < 1 \), then \( {r}^{n} \rightarrow 0 \) as \( n \rightarrow \infty \) . (Given \( \varepsilon > 0 \), first choose \( \nu \) such that \( 1/\nu < \varepsilon \) . Then choose \( k \) such that \( \left. {{\left| r\right| }^{-1} > 1 + {k}^{-1}\text{. }}\right) \) .3 Prove that if \( r > 1 \), then \( {r}^{n} \rightarrow \infty \) as \( n \rightarrow \infty \) . .4 Prove that if \( a > 1 \), then \( {\log }_{a}n \rightarrow \infty \) as \( n \rightarrow \infty \) . .5 Prove that if \( r = \mathop{\lim }\limits_{{n \rightarrow \infty }}{a}_{n} \), then \( r = \mathop{\lim }\limits_{{k \rightarrow \infty }}{a}_{{n}_{k}} \) for any subsequence \( {\left( {a}_{{n}_{k}}\right) }_{k = 1}^{\infty } \) of \( \left( {a}_{n}\right) \) . .6 Let \( \left( {a}_{n}\right) \) be a sequence of real numbers such that the subsequences \( {\left( {a}_{2n}\right) }_{n = 1}^{\infty } \) and \( {\left( {a}_{{2n} + 1}\right) }_{n = 1}^{\infty } \) both converge to the limit \( l \) . Prove that \( \left( {a}_{n}\right) \) converges to \( l \) . .7 Let \( \left( {a}_{n}\right) \) be a sequence in \( \mathbf{R} \) . Prove that if the three subsequences \( \left( {a}_{2n}\right) ,\left( {a}_{{2n} + 1}\right) \), and \( \left( {a}_{3n}\right) \) are convergent, then so is \( \left( {a}_{n}\right) \) . .8 Give an example of a sequence \( \left( {a}_{n}\right) \) of real numbers with the following properties. (i) \( \left( {a}_{n}\right) \) is not convergent; (ii) for each \( k \geq 2 \) the subsequence \( {\left( {a}_{kn}\right) }_{n = 1}^{\infty } \) is convergent. (Split your definition of \( {a}_{n} \) into two cases - one when \( n \) is prime, the other when \( n \) is composite.) When we apply notions such as bounded above, supremum, and infimum to a sequence \( \left( {s}_{n}\right) \) of real numbers, we are really applying them to the set \( \left\{ {{s}_{n} : n \geq 1}\right\} \) of terms of the sequence. Thus the supremum (respectively, infimum) of a majorised (respectively, minorised) sequence \( \left( {s}_{n}\right) \) is denoted by \( \mathop{\sup }\limits_{{n \geq 1}}{s}_{n} \), or just \( \sup {s}_{n} \) (respectively, \( \mathop{\inf }\limits_{{n \geq 1}}{s}_{n} \), or just \( \inf {s}_{n} \) ). The next result, known as the monotone sequence principle, is a powerful tool for proving the existence of limits. (1.2.4) Proposition. An increasing majorised sequence of real numbers converges to its least upper bound; a decreasing minorised sequence of real numbers converges to its greatest lower bound. Proof. Let \( \left( {s}_{n}\right) \) be an increasing majorised sequence of real numbers, and \( s \) its least upper bound. For each \( \varepsilon > 0 \), since \( s - \varepsilon \) is not an upper bound of \( \left( {s}_{n}\right) \), there exists \( N \) such that \( {s}_{N} > s - \varepsilon \) . But \( \left( {s}_{n}\right) \) is both increasing and bounded above by \( s \) ; so for all \( n \geq N \) we have \( s - \varepsilon < {s}_{n} \leq s \) and therefore \( \left| {s - {s}_{n}}\right| < \varepsilon \) . Since \( \varepsilon > 0 \) is arbitrary, it follows that \( {s}_{n} \rightarrow s \) as \( n \rightarrow \infty \) . The case of a decreasing minorised sequence is left as an exercise. ## (1.2.5) Exercises .1 Prove the second part of the last proposition in two ways. .2 Prove that an increasing sequence of nonnegative real numbers diverges to infinity if and only if it is not bounded above. .3 Let \( a > 1 \) and \( x > 0 \) . Prove that there exists an integer \( m \) such that \( {a}^{m} \leq x < {a}^{m + 1} \) . (First take \( x \geq 1 \), and consider the sequence \( \left. {{\left( {a}^{n}\right) }_{n = 0}^{\infty }\text{.}}\right) \) .4 Discuss the convergence of the sequence \( \left( {a}_{n}\right) \) defined by \( {a}_{n + 1} = \) \( \sqrt{r{a}_{n}} \), where \( {a}_{1} \) and \( r \) are positive numbers. .5 Prove that if \( 0 < a \) and \( k \in \mathbf{N} \), then \( \mathop{\lim }\limits_{{n \rightarrow \infty }}\sqrt[{n + k}]{a} = 1 \) . (First consider the case where \( k = 0 \) and \( 0 < a < 1 \) . Apply the monotone sequence principle to show that the sequence \( {\left( \sqrt[n]{a}\right) }_{n = 1}^{\infty } \) converges to a limit \( l \) . By considering the subsequence \( \left( {\sqrt[{2n}]{a})\text{, show that}\sqrt{l} = l\text{.}}\right) \) .6 Prove that if \( \left( {a}_{n}\right) \) is a sequence of positive numbers such that \[ l = \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{{a}_{n + 1}}{{a}_{n}} \] exists, then \( \mathop{\lim }\limits_{{n \rightarrow \infty }}\sqrt[n]{{a}_{n}} = l \) . By considering the sequence \[ 1, a,{ab},{a}^{2}b,{a}^{2}{b}^{2},{a}^{3}{b}^{2},{a}^{3}{b}^{3},\ldots , \] where \( a, b \) are distinct positive numbers, show that the converse is false. .7 Prove that if \( n \geq 2 \), then \( {\left( n + 1\right) }^{n} \leq {n}^{n + 1} \) . Use this to show that \( l = \mathop{\lim }\limits_{{n \rightarrow \infty }}\sqrt[n]{n} \) exists. By considering the subsequence \( {\left( \sqrt[{2n}]{2n}\right) }_{n = 1}^{\infty } \), prove that \( l = 1 \) . Hence show that if \( a > 1 \), then \( \mathop{\lim }\limits_{{n \rightarrow \infty }}\left( {{n}^{-1}{\log }_{a}n}\right) = 0 \) . .8 Prove that the sequence \( {\left( {\left( 1 + {n}^{-1}\right) }^{n}\right) }_{n = 1}^{\infty } \) is convergent. (An interesting proof of this result, based on the well-known inequality involving arithmetic and geometric means, is found in [32].) .9 Let \( \left( {a}_{n}\right) \) be a sequence of real numbers. If \( \left( {a}_{n}\right) \) is bounded above, then its upper limit, or limit superior, is defined to be \[ \lim \sup {a}_{n} = \mathop{\inf }\limits_{{n \geq 1}}\sup \left\{ {{a}_{n},{a}_{n + 1},{a}_{n + 2},\ldots }\right\} \] if the infimum on the right exists. Prove that a real number \( s \) equals \( \lim \sup {a}_{n} \) if and only if for each \( \varepsilon > 0 \) , \( - {a}_{n} < s + \varepsilon \) for all sufficiently large \( n \), and - \( {a}_{n} > s - \varepsilon \) for infinitely many values of \( n \) . Prove also that \[ \lim \sup {a}_{n} = \mathop{\lim }\limits_{{n \rightarrow \infty }}\sup \left\{ {{a}_{n},{a}_{n + 1},{a}_{n + 2},\ldots }\right\} . \] .10 If \( \left( {a}_{n}\right) \) is bounded below, then its lower limit, or limit inferior, is defined to be \[ \lim \inf {a}_{n} = \mathop{\sup }\limits_{{n \geq 1}}\inf \left\{ {{a}_{n},{a}_{n + 1},{a}_{n + 2},\ldots }\right\} \] if the supremum on the right exists. Establish necessary and sufficient conditions for a real number \( l \) to equal \( \liminf {a}_{n} \) . .11 Prove that \( {a}_{n} \rightarrow a \in \mathbf{R} \) as \( n \rightarrow \infty \) if and only if \[ \liminf {a}_{n} = a = \limsup {a}_{n}. \] A sequence \( {\left( {S}_{n}\right) }_{n = 1}^{\infty } \) of subsets of \( \mathbf{R} \) is said to be nested, or descending, if \( {S}_{1} \supset {S}_{2} \supset {S}_{3} \supset \cd

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varepsilon \) for all sufficiently large \( n \), and - \( {a}_{n} > s - \varepsilon \) for infinitely many values of \( n \) . Prove also that \[ \lim \sup {a}_{n} = \mathop{\lim }\limits_{{n \rightarrow \infty }}\sup \left\{ {{a}_{n},{a}_{n + 1},{a}_{n + 2},\ldots }\right\} . \] .10 If \( \left( {a}_{n}\right) \) is bounded below, then its lower limit, or limit inferior, is defined to be \[ \lim \inf {a}_{n} = \mathop{\sup }\limits_{{n \geq 1}}\inf \left\{ {{a}_{n},{a}_{n + 1},{a}_{n + 2},\ldots }\right\} \] if the supremum on the right exists. Establish necessary and sufficient conditions for a real number \( l \) to equal \( \liminf {a}_{n} \) . .11 Prove that \( {a}_{n} \rightarrow a \in \mathbf{R} \) as \( n \rightarrow \infty \) if and only if \[ \liminf {a}_{n} = a = \limsup {a}_{n}. \] A sequence \( {\left( {S}_{n}\right) }_{n = 1}^{\infty } \) of subsets of \( \mathbf{R} \) is said to be nested, or descending, if \( {S}_{1} \supset {S}_{2} \supset {S}_{3} \supset \cdots \) . We make good use of the following nested intervals principle. (1.2.6) Proposition. The intersection of a nested sequence of closed intervals in \( \mathbf{R} \) is nonempty. Proof. Let \( \left( \left\lbrack {{a}_{n},{b}_{n}}\right\rbrack \right) \) be a nested sequence of closed intervals in \( \mathbf{R} \) . Then \[ {a}_{1} \leq {a}_{n} \leq {a}_{n + 1} \leq {b}_{n + 1} \leq {b}_{n} \leq {b}_{1} \] (1) for each \( n \) . By Proposition (1.2.4), \( \left( {a}_{n}\right) \) converges to its least upper bound \( a \), and \( \left( {b}_{n}\right) \) converges to its greatest lower bound \( b \) . It follows from the inequalities (1) and Exercise (1.2.1: 5) that \( a \leq b \) . So for each \( n,{a}_{n} \leq a \leq \) \( b \leq {b}_{n} \) and therefore \( a \in \left\lbrack {{a}_{n},{b}_{n}}\right\rbrack \) . The following elementary lemma leads to simple proofs of several important results in analysis. (1.2.7) Lemma. If \( \left( {a}_{n}\right) \) is a sequence of real numbers, then at least one of the following holds. (i) \( \left( {a}_{n}\right) \) has a constant subsequence; (ii) \( \left( {a}_{n}\right) \) has a strictly increasing subsequence; (iii) \( \left( {a}_{n}\right) \) has a strictly decreasing subsequence. Proof. Suppose that \( \left( {a}_{n}\right) \) contains no constant subsequence, and consider the set \[ S = \left\{ {n \in {\mathbf{N}}^{ + } : \forall k \geq n\left( {{a}_{n} \geq {a}_{k}}\right) }\right\} . \] If \( S \) is bounded, then there exists \( N \) such that \[ \forall n \geq N\exists k > n\left( {{a}_{k} > {a}_{n}}\right) , \] and a simple inductive construction produces positive integers \( N \leq {n}_{1} < \) \( {n}_{2} < \cdots \) such that \( {a}_{{n}_{k + 1}} > {a}_{{n}_{k}} \) for each \( k \) . If, on the other hand, \( S \) is unbounded, then we can compute \( {n}_{1} < {n}_{2} < \cdots \) such that \( {a}_{{n}_{k}} \geq {a}_{{n}_{k + 1}} \) for each \( k \) . In that case, since \( {\left( {a}_{{n}_{k}}\right) }_{k = 1}^{\infty } \) contains no constant subsequence, for each \( k \) there exists \( j > k \) such that \( {a}_{{n}_{k}} > {a}_{{n}_{j}} \) ; it is now straightforward to construct a strictly decreasing subsequence of \( \left( {a}_{{n}_{k}}\right) \) . (1.2.8) Corollary. A bounded sequence of real numbers has a convergent subsequence. Proof. This follows from Lemma (1.2.7) and the monotone sequence principle. A sequence \( \left( {a}_{n}\right) \) of real numbers is called a Cauchy sequence if for each \( \varepsilon > 0 \) there exists a positive integer \( N \), depending on \( \varepsilon \), such that \( \left| {{a}_{m} - {a}_{n}}\right| \leq \) \( \varepsilon \) for all \( m, n \geq N \) . ## (1.2.9) Exercises .1 Prove that a convergent sequence of real numbers is a Cauchy sequence. .2 Prove that a Cauchy sequence is bounded. .3 Prove that if a Cauchy sequence \( \left( {a}_{n}\right) \) has a subsequence that converges to a limit \( a \in \mathbf{R} \), then \( \left( {a}_{n}\right) \) converges to \( a \) . .4 Let \( \left( {a}_{n}\right) \) be a bounded sequence each of whose convergent subsequences converges to the same limit. Prove that \( \left( {a}_{n}\right) \) converges to that limit. (cf. Exercises (1.2.3: 6 and 7). By Corollary (1.2.8), there is a subsequence \( \left( {a}_{{n}_{k}}\right) \) that converges to a limit \( l \) . Suppose that \( \left( {a}_{n}\right) \) does not converge to \( l \), and derive a contradiction.) One of the most important results in convergence theory says that not only does a Cauchy sequence of real numbers appear to converge, in that its terms get closer and closer to each other as their indices increase, but it actually does converge. A subset \( S \) of \( \mathbf{R} \) is said to be complete if each Cauchy sequence in \( S \) converges to a limit that belongs to \( S \) . (1.2.10) Theorem. R is complete. Proof. Let \( \left( {a}_{n}\right) \) be a Cauchy sequence in \( \mathbf{R} \) . Then \( \left( {a}_{n}\right) \) is bounded, by Exercise (1.2.9:2). It follows from Corollary (1.2.8) that \( \left( {a}_{n}\right) \) has a convergent subsequence; so \( \left( {a}_{n}\right) \) converges, by Exercise (1.2.9:3). \( ▱ \) ## (1.2.11) Exercises .1 Find an alternative proof of the completeness of \( \mathbf{R} \) . (Given a Cauchy sequence \( \left( {a}_{n}\right) \) in \( \mathbf{R} \), consider \( \liminf {a}_{n} \) .) .2 Show that if, in the system of axioms for \( \mathbf{R} \), the least-upper-bound principle is replaced by the Axiom of Archimedes (Exercise (1.1.1:16)), then the nested intervals principle is equivalent to the completeness of \( \mathbf{R} \) . Can you spot where you have used the Axiom of Archimedes? .3 Under the conditions of the preceding exercise, show that the least-upper-bound principle follows from the completeness of \( \mathbf{R} \) . (Assuming that \( \mathbf{R} \) is complete, consider a nonempty majorised subset \( S \) of R. Choose \( {s}_{1} \in S \) and \( {b}_{1} \in B \), where \( B \) is the set of upper bounds of \( S \) . Construct a sequence \( \left( {s}_{n}\right) \) in \( S \) and a sequence \( \left( {b}_{n}\right) \) in \( B \) such that \[ {s}_{n} \leq {s}_{n + 1} \leq {b}_{n + 1} \leq {b}_{n} \] and \[ 0 \leq {b}_{n + 1} - {s}_{n + 1} \leq \frac{1}{2}\left( {{b}_{n} - {s}_{n}}\right) \] Prove that \( 0 \leq {b}_{n} - {b}_{m} \leq {2}^{-n + 2}\left( {{b}_{1} - {s}_{1}}\right) \) whenever \( m \geq n \), that \( \left( {s}_{n}\right) \) and \( \left( {b}_{n}\right) \) converge to the same limit \( b \), and that \( b = \sup S \) .) .4 Prove Cantor’s Theorem: if \( \left( {a}_{n}\right) \) is a sequence of real numbers, then in any closed interval of \( \mathbf{R} \) with positive length there exists a real number \( x \) such that \( x \neq {a}_{n} \) for each \( n \) . (For each \( x \in \mathbf{R} \) and each nonempty \( S \subset \mathbf{R} \), define the distance from \( x \) to \( S \) to be the real number \[ \rho \left( {x, S}\right) = \inf \{ \left| {x - s}\right| : s \in S\} . \] First prove the following lemma. If \( I = \left\lbrack {a, b}\right\rbrack \) is a closed interval with positive length, and \( {J}_{1},{J}_{2},{J}_{3} \) are the left, middle, and right closed thirds of \( I \), then for each real number \( x \) either \( \rho \left( {x,{J}_{1}}\right) > 0 \) or \( \rho \left( {x,{J}_{3}}\right) > 0 \) . Use this lemma to construct an appropriate nested sequence of closed intervals. This argument is a refined version of the "diagonal argument" first used by Cantor. An interesting analysis of Cantor's proof, and of the misinterpretation of that proof over the years, is found in [19].) ## .5 Prove that \( \mathbf{R} \smallsetminus \mathbf{Q} \) is order dense in \( \mathbf{R} \) . The study of infinite series, a major part of analysis in the eighteenth and nineteenth centuries (see [27]), still provides interesting illustrations of the completeness of \( \mathbf{R} \) . Let \( {\left( {a}_{n}\right) }_{n = 1}^{\infty } \) be a sequence of real numbers. The real number \[ {s}_{k} = \mathop{\sum }\limits_{{n = 1}}^{k}{a}_{n} \] is called the \( k \) th partial sum of the series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n} \) . Formally, we define the series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n} \) with \( n \) th term \( {a}_{n} \) to be the sequence \( \left( {{s}_{1},{s}_{2},\ldots }\right) \) of its partial sums. The sum of that series is the limit \( s \) of the sequence \( \left( {s}_{n}\right) \), if that limit exists, in which case we say that the series is convergent, or that it converges to \( s \), and we write \[ \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n} = s \] We use analogous notations and definitions for the series associated with a family \( {\left( {a}_{n}\right) }_{n = \nu }^{\infty } \) of real numbers indexed by \( \{ n \in \mathbf{Z} : n \geq \nu \} \), where \( \nu \) is an integer, and for the series \( \mathop{\sum }\limits_{{n = - \infty }}^{\infty }{a}_{n} \) associated with a family \( {\left( {a}_{n}\right) }_{n \in \mathbf{Z}} \) indexed by \( \mathbf{Z} \) . We commonly write \( \sum {a}_{n} \) for the series \( \mathop{\sum }\limits_{{n = \nu }}^{\infty }{a}_{n} \), when it is clear that the indexing of the terms of the series starts with \( \nu \) . The completeness of \( \mathbf{R} \) is used in the justification of various tests for the convergence of infinite series. These tests are useful because they enable us to prove certain series convergent without finding explicit values for their sums. For example, a number of convergence tests easily show that the series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{n}^{-2} \) is convergent; but it is considerably harder to show that the sum of this series is actually \( {\pi }^{2}/6 \) (Exercise (5.2.12:7); see also [31]). We begin with the comparison test. (1.2.12) Proposition. If \( \mathop{\sum }\limits_{{n

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or the series \( \mathop{\sum }\limits_{{n = - \infty }}^{\infty }{a}_{n} \) associated with a family \( {\left( {a}_{n}\right) }_{n \in \mathbf{Z}} \) indexed by \( \mathbf{Z} \) . We commonly write \( \sum {a}_{n} \) for the series \( \mathop{\sum }\limits_{{n = \nu }}^{\infty }{a}_{n} \), when it is clear that the indexing of the terms of the series starts with \( \nu \) . The completeness of \( \mathbf{R} \) is used in the justification of various tests for the convergence of infinite series. These tests are useful because they enable us to prove certain series convergent without finding explicit values for their sums. For example, a number of convergence tests easily show that the series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{n}^{-2} \) is convergent; but it is considerably harder to show that the sum of this series is actually \( {\pi }^{2}/6 \) (Exercise (5.2.12:7); see also [31]). We begin with the comparison test. (1.2.12) Proposition. If \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{b}_{n} \) is a convergent series of nonnegative terms, and if \( 0 \leq {a}_{n} \leq {b}_{n} \) for each \( n \), then \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n} \) converges. Proof. Let \( b \) be the sum of the series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{b}_{n} \) . Then for each \( N \) we have \[ \mathop{\sum }\limits_{{n = 1}}^{N}{a}_{n} \leq \mathop{\sum }\limits_{{n = 1}}^{{N + 1}}{a}_{n} \leq \mathop{\sum }\limits_{{n = 1}}^{{N + 1}}{b}_{n} \leq b \] so the partial sums of \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n} \) form an increasing majorised sequence. It follows from the monotone sequence principle that \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n} \) converges. (1.2.13) Proposition. If \( \left( {a}_{n}\right) \) is a decreasing sequence of positive numbers converging to 0, then the alternating series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{\left( -1\right) }^{n + 1}{a}_{n} \) converges (Leibniz's alternating series test). Proof. For each \( k \) let \[ {s}_{k} = \mathop{\sum }\limits_{{n = 1}}^{k}{\left( -1\right) }^{n + 1}{a}_{n} \] Then \[ {s}_{{2k} + 2} - {s}_{2k} = {a}_{{2k} + 1} - {a}_{{2k} + 2} \geq 0 \] and \[ {a}_{1} - {s}_{2k} = \left( {{a}_{2} - {a}_{3}}\right) + \ldots + \left( {{a}_{{2k} - 2} - {a}_{{2k} - 1}}\right) + {a}_{2k} \geq 0. \] So the sequence \( {\left( {s}_{2k}\right) }_{k = 1}^{\infty } \) is increasing and bounded above; whence, by the monotone sequence principle, it converges to its least upper bound \( s \) . Now, \[ \left| {s - {s}_{{2m} + 1}}\right| = \left| {s - {s}_{2m} - {a}_{{2m} + 1}}\right| \leq \left| {s - {s}_{2m}}\right| + {a}_{{2m} + 1}. \] Also, both \( \left| {s - {s}_{2m}}\right| \) and \( {a}_{{2m} + 1} \) converge to 0 as \( m \rightarrow \infty \) . It follows that if \( \varepsilon > 0 \), then \( \left| {s - {s}_{2m}}\right| < \varepsilon \) and \( \left| {s - {s}_{{2m} + 1}}\right| < \varepsilon \) for all sufficiently large \( m \) . Hence \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{\left( -1\right) }^{n + 1}{a}_{n} \) converges to \( s \), by Exercise (1.2.3: 6). ## (1.2.14) Exercises .1 Prove that if the series \( \sum {a}_{n} \) converges, then \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{a}_{n} = 0 \) . By considering \( \mathop{\sum }\limits_{{n = 1}}^{\infty }1/\sqrt{n} \), or otherwise, show that the converse is false. .2 Prove the comparison test using the completeness of \( \mathbf{R} \), instead of the least-upper-bound principle. .3 A series of nonnegative terms is said to diverge if the corresponding sequence \( \left( {s}_{n}\right) \) of partial sums diverges to infinity. Prove the limit comparison test: If \( \left( {a}_{n}\right) \) and \( \left( {b}_{n}\right) \) are sequences of positive numbers such that \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{{a}_{n}}{{b}_{n}} = l > 0 \] then either \( \sum {a}_{n} \) and \( \sum {b}_{n} \) both converge or else they both diverge. .4 Prove that if \( \left| r\right| < 1 \), then the geometric series \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{r}^{n} \) converges and has sum \( 1/\left( {1 - r}\right) \) . What happens to the series if \( \left| r\right| > 1 \) ? .5 Let \( b \geq 2 \) be an integer, and \( x \in \left\lbrack {0,1}\right\rbrack \) . Show that there exists a sequence \( \left( {a}_{n}\right) \) of integers such that (i) \( 0 \leq {a}_{n} < b \) for each \( n \), and (ii) \( x = \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}{b}^{-n} \) . Show that this sequence \( \left( {a}_{n}\right) \) is uniquely determined by \( x \) unless there exist \( k, n \in \mathbf{N} \) such that \( x = k{b}^{-n} \), in which case there are exactly two such sequences. Conversely, show that if \( \left( {a}_{n}\right) \) is a sequence of integers satisfying (i), then \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}{b}^{-n} \) converges to a sum \( x \) in \( \left\lbrack {0,1}\right\rbrack \) . (The series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}{b}^{-n} \) is called the \( b \) -ary expansion of \( x \), or the expansion of \( x \) relative to the base \( b \) . If \( b = 2 \), the series is the binary expansion of \( x \), and if \( b = {10} \), it is the decimal expansion.) .6 Prove that (i) \( \mathop{\sum }\limits_{{n = 1}}^{\infty }1/{n}^{p} \) is divergent if \( p \leq 1 \) ; (ii) \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{\left( -1\right) }^{n}/n \) is convergent. (For (i), first prove the divergence of \( \mathop{\sum }\limits_{{n = 1}}^{\infty }1/n \) by considering the partial sums \( \mathop{\sum }\limits_{{n = 1}}^{{2}^{N}}1/n \) for \( N = 1,2,\ldots \) ) .7 Prove that the series \[ \frac{1}{9} + \frac{1}{19} + \frac{1}{29} + \cdots + \frac{1}{89} + \frac{1}{90} + \frac{1}{91} + \cdots + \frac{1}{99} + \frac{1}{109} + \frac{1}{119} + \cdots , \] where each term contains the digit 9 , diverges; and that the series \[ 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{8} + \frac{1}{10} + \cdots + \frac{1}{18} + \frac{1}{20} + \cdots , \] where no term contains the digit 9 , converges. (Thus the divergent series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }1/n \) can be turned into a convergent one by weeding out all the terms that contain the digit 9 . For a discussion of this and related matters, see [3].) .8 Prove that if \( p \geq 2 \), then the series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }1/{n}^{p} \) is convergent. .9 Prove d’Alembert’s ratio test: let \( \sum {a}_{n} \) be a series of positive terms such that \[ l = \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{{a}_{n + 1}}{{a}_{n}} \] exists; then \( \sum {a}_{n} \) converges if \( l < 1 \), and diverges if \( l > 1 \) . (In the first case, choose \( r \in \left( {l,1}\right) \) and \( N \) such that \( 0 < {a}_{n + 1} < r{a}_{n} \) for all \( n \geq N \) .) Give examples where \( l = 1 \) and (i) \( \sum {a}_{n} \) converges,(ii) \( \sum {a}_{n} \) diverges. .10 Prove that \( \mathop{\sum }\limits_{{n = 0}}^{\infty }1/n \) ! converges and has sum \( < 3 \) . Show also that \[ \mathop{\sum }\limits_{{k = 0}}^{n}\frac{1}{k!} - \frac{3}{2n} < {\left( 1 + \frac{1}{n}\right) }^{n} < \mathop{\sum }\limits_{{k = 0}}^{n}\frac{1}{k!} \] for all \( n \geq 3 \), and hence prove that \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{\left( 1 + {n}^{-1}\right) }^{n} = \mathop{\sum }\limits_{{n = 0}}^{\infty }1/n! \) . .11 Prove Cauchy’s root test: let \( \sum {a}_{n} \) be a series of positive terms, and \[ l = \limsup \sqrt[n]{{a}_{n}} \] then \( \sum {a}_{n} \) converges if \( l < 1 \), and diverges if \( l > 1 \) . .12 Discuss the convergence of the series \[ \frac{1}{2} + \frac{1}{3} + \frac{1}{{2}^{2}} + \frac{1}{{3}^{2}} + \frac{1}{{2}^{3}} + \frac{1}{{3}^{3}} + \cdots . \] What does this series and Exercise (1.2.5: 6) tell you about the relative strengths of the ratio test and the root test? .13 Let \( \left( {a}_{n}\right) \) be a decreasing sequence of nonnegative real numbers, and for each \( N \) let \[ {s}_{N} = \mathop{\sum }\limits_{{n = 1}}^{N}{a}_{n},{t}_{N} = \mathop{\sum }\limits_{{n = 1}}^{N}{2}^{n}{a}_{{2}^{n}}. \] Show that (i) if \( m \leq {2}^{N} \), then \( {s}_{m} \leq {t}_{N} \), and (ii) if \( m \geq {2}^{N} \), then \( {s}_{m} \geq \frac{1}{2}{t}_{N} \) . Hence prove that \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n} \) converges if and only if \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{2}^{n}{a}_{{2}^{n}} \) converges. .14 Use the preceding exercise to show that \( \mathop{\sum }\limits_{{n = 1}}^{\infty }1/{n}^{p} \) converges if and only if \( p > 1 \) (cf. Exercises (1.2.14: 6 and 8)). .15 Let \( {\left( {a}_{n}\right) }_{n = 0}^{\infty } \) and \( {\left( {b}_{n}\right) }_{n = 0}^{\infty } \) be sequences of real numbers, and for each \( N \) write \( {S}_{N} = \mathop{\sum }\limits_{{n = 0}}^{N}{a}_{n} \) . Show that if \( k > j \), then \[ \mathop{\sum }\limits_{{n = j}}^{k}{a}_{n}{b}_{n} = \mathop{\sum }\limits_{{n = j}}^{{k - 1}}{S}_{n}\left( {{b}_{n} - {b}_{n + 1}}\right) + {S}_{k}{b}_{k} - {S}_{j - 1}{b}_{j}. \] Now suppose that (i) there exists \( M > 0 \) such that \( \left| {S}_{n}\right| \leq M \) for all \( n \) , (ii) \( {b}_{n} \geq {b}_{n + 1} \) for each \( n \), and (iii) \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{b}_{n} = 0 \) . Prove that if \( k > j \), then \( \left| {\mathop{\sum }\limits_{{n = j}}^{k}{a}_{n}{b}_{n}}\right| < {2M}{b}_{j} \), and hence that \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{b}_{n} \) converges. Use this result to give another proof of Leibniz's alternating series test. A series \( \sum {a}_{n} \) of real numbers is said to be absolutely convergent if \( \sum \left| {a}_{n}\right| \) is convergent. (1.2.15) Proposition. An absolutely convergent series is convergent. Proof. Let \( \sum {a}_{n} \) be absolutely convergent. Since the partial sums of \( \sum \left| {a}_{n}\right| \) form a Cauchy

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{n}{b}_{n} = \mathop{\sum }\limits_{{n = j}}^{{k - 1}}{S}_{n}\left( {{b}_{n} - {b}_{n + 1}}\right) + {S}_{k}{b}_{k} - {S}_{j - 1}{b}_{j}. \] Now suppose that (i) there exists \( M > 0 \) such that \( \left| {S}_{n}\right| \leq M \) for all \( n \) , (ii) \( {b}_{n} \geq {b}_{n + 1} \) for each \( n \), and (iii) \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{b}_{n} = 0 \) . Prove that if \( k > j \), then \( \left| {\mathop{\sum }\limits_{{n = j}}^{k}{a}_{n}{b}_{n}}\right| < {2M}{b}_{j} \), and hence that \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{b}_{n} \) converges. Use this result to give another proof of Leibniz's alternating series test. A series \( \sum {a}_{n} \) of real numbers is said to be absolutely convergent if \( \sum \left| {a}_{n}\right| \) is convergent. (1.2.15) Proposition. An absolutely convergent series is convergent. Proof. Let \( \sum {a}_{n} \) be absolutely convergent. Since the partial sums of \( \sum \left| {a}_{n}\right| \) form a Cauchy sequence, for each \( \varepsilon > 0 \) there exists \( N \) such that \[ \left| {\mathop{\sum }\limits_{{n = 1}}^{k}\left| {a}_{n}\right| - \mathop{\sum }\limits_{{n = 1}}^{j}\left| {a}_{n}\right| }\right| = \mathop{\sum }\limits_{{n = j}}^{k}\left| {a}_{n}\right| < \varepsilon \] whenever \( k > j \geq N \) . For such \( j \) and \( k \) we have \[ \left| {\mathop{\sum }\limits_{{n = 1}}^{k}{a}_{n} - \mathop{\sum }\limits_{{n = 1}}^{j}{a}_{n}}\right| = \left| {\mathop{\sum }\limits_{{n = j}}^{k}{a}_{n}}\right| \leq \mathop{\sum }\limits_{{n = j}}^{k}\left| {a}_{n}\right| < \varepsilon . \] Thus the partial sums of \( \sum {a}_{n} \) form a Cauchy sequence; whence \( \sum {a}_{n} \) is convergent, by the completeness of \( \mathbf{R} \) . The case \( p = 1 \) of Example (1.2.14:6) shows that the converse of Proposition (1.2.15) is false. By a power series we mean a series of the form \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n} \), where the coefficients \( {a}_{n} \in \mathbf{R} \) . Such a series always converges for \( x = 0 \), but it may converge for nonzero values of \( x \) . Its radius of convergence is defined to be \[ \sup \left\{ {r \geq 0 : \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n}\text{ converges whenever }\left| x\right| \leq r}\right\} \] if this supremum exists, and \( \infty \) otherwise; and its interval of convergence is the largest interval \( I \) such that the power series converges for all \( x \in I \) . It is an immediate consequence of Exercise (1.2.16:10) that every power series has both a radius and an interval of convergence. ## (1.2.16) Exercises .1 Find an alternative proof of Proposition (1.2.15). .2 Prove that the series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }n{r}^{n} \) converges absolutely if \( - 1 < r < 1 \) . .3 Let \( \sum {a}_{n},\sum {b}_{n} \) be convergent series of nonnegative terms, with sums \( a, b \), respectively, and let \[ {u}_{n} = {a}_{1}{b}_{n} + {a}_{2}{b}_{n - 1} + \cdots + {a}_{n - 1}{b}_{2} + {a}_{n}{b}_{1}. \] Prove that \[ \mathop{\sum }\limits_{{n = 1}}^{N}{u}_{n} \leq \left( {\mathop{\sum }\limits_{{n = 1}}^{N}{a}_{n}}\right) \left( {\mathop{\sum }\limits_{{n = 1}}^{N}{b}_{n}}\right) \leq \mathop{\sum }\limits_{{n = 1}}^{N}{u}_{2n} \] and hence that \( \sum {u}_{n} \) converges to the sum ab (Cauchy’s theorem on the multiplication of series). Extend this result to the case where the terms \( {a}_{n} \) may not be nonnegative but \( \sum {a}_{n} \) is absolutely convergent. (Writing \[ b = \mathop{\sum }\limits_{{n = 1}}^{\infty }{b}_{n} \] \[ {\beta }_{k} = b - \mathop{\sum }\limits_{{n = 1}}^{k}{b}_{n} \] show that \[ \mathop{\sum }\limits_{{n = 1}}^{N}{u}_{n} = b\mathop{\sum }\limits_{{n = 1}}^{N}{a}_{n} + \mathop{\sum }\limits_{{k = 0}}^{N}{a}_{k}{\beta }_{N - k} \] and hence that \( \mathop{\sum }\limits_{{k = 0}}^{N}{a}_{k}{\beta }_{N - k} \rightarrow 0 \) as \( N \rightarrow \infty \) .) .4 Show that \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{\left( -1\right) }^{n}/\sqrt{n + 1} \) converges, but that the product (as in the preceding exercise) of this series with itself does not converge. .5 Prove that the exponential series \[ \exp \left( x\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{{x}^{n}}{n!} \] converges absolutely for all \( x \in \mathbf{R} \) . Then prove that \[ \exp \left( {x + y}\right) = \exp \left( x\right) \exp \left( y\right) \] .6 Prove that \( \exp \left( x\right) = {\mathrm{e}}^{x} \), where \( \mathrm{e} = \exp \left( 1\right) \) . (Use Exercise (1.1.4:5).) Show that \[ 0 < \mathrm{e} - \mathop{\sum }\limits_{{n = 0}}^{N}\frac{1}{n!} < \frac{3}{\left( {N + 1}\right) !} \] for each \( N \), and hence calculate e with an error at most \( {10}^{-6} \) . .7 Prove that \( \mathrm{e} \) is irrational. (Suppose that \( \mathrm{e} = p/q \), where \( p \) and \( q \) are positive integers. Choose \( N > \max \{ q,3\} \), show that \( N!\mathop{\sum }\limits_{{n = N + 1}}^{\infty }1/n! \) is an integer, and use the inequality from the preceding exercise to deduce a contradiction.) .8 Show that \[ {\mathrm{e}}^{x} = \mathop{\lim }\limits_{{n \rightarrow \infty }}{\left( 1 + \frac{x}{n}\right) }^{n} \] for each \( x \in \mathbf{R} \) . (First take \( x > 0 \) . Expand \( {s}_{n} = {\left( 1 + x/n\right) }^{n} \) using the binomial theorem, and use the monotone sequence principle.) .9 For each \( n \) define \[ {\gamma }_{n} = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} - \log n \] Show that \[ \mathrm{e} < {\left( 1 + \frac{1}{n}\right) }^{n + 1} < {\mathrm{e}}^{1 + {n}^{-1}} \] for each \( n \), and hence that the sequence \( \left( {\gamma }_{n}\right) \) is decreasing and bounded below. It follows from the monotone sequence principle that Euler's constant \[ \gamma = \mathop{\lim }\limits_{{n \rightarrow \infty }}{\gamma }_{n} \] exists. Show that \[ \mathop{\sum }\limits_{{n = 1}}^{N}\frac{{\left( -1\right) }^{n + 1}}{n} = {\gamma }_{2N} - {\gamma }_{N} + \log 2 \] and hence that \[ \mathop{\sum }\limits_{{n = 1}}^{N}\frac{{\left( -1\right) }^{n + 1}}{n} = \log 2 \] (cf. Exercise (1.2.14: 6).) .10 Let \( r > 0 \) . Prove that if \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n} \) converges for \( x = r \), then it converges absolutely whenever \( \left| x\right| < r \) ; and that if this power series diverges for \( x = r \), then it diverges whenever \( \left| x\right| > r \) . (For the first part, show that there exists \( M > 0 \) such that \( \left| {{a}_{n}{x}^{n}}\right| \leq M{\left| x/r\right| }^{n} \) for all \( n \) .) .11 Find the radius of convergence and the interval of convergence for \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{x}^{n} \) . .12 Find the radius of convergence and the interval of convergence for \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{\left( -1\right) }^{n}{x}^{n}/n \) . .13 Suppose that \( {a}_{n} \neq 0 \) for all \( n \), and that \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\left| \frac{{a}_{n + 1}}{{a}_{n}}\right| = l \] Show that if \( l = 0 \), then \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n} \) converges for all \( x \in \mathbf{R} \) ; and that if \( l \neq 0 \), then the series has radius of convergence \( 1/l \) . .14 Let \( {\left( {a}_{n}\right) }_{n = 0}^{\infty } \) be a bounded sequence of real numbers, and let \[ l = \lim \sup \sqrt[n]{\left| {a}_{n}\right| }. \] Show that if \( l = 0 \), then \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n} \) converges for all \( x \in \mathbf{R} \) ; and that if \( l \neq 0 \), then the series has radius of convergence \( 1/l \) . .15 Prove that if \( {\left( \sqrt[n]{\left| {a}_{n}\right| }\right) }_{n = 0}^{\infty } \) is an unbounded sequence, then the power series \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n} \) only converges for \( x = 0 \) . .16 Prove that the power series \[ \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n},\mathop{\sum }\limits_{{n = 1}}^{\infty }n{a}_{n}{x}^{n - 1}\text{, and }\mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{{a}_{n}}{n + 1}{x}^{n + 1} \] have the same radius of convergence. Need they have the same interval of convergence? By a rearrangement of an infinite series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n} \) we mean a series of the form \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{f\left( n\right) } \) where \( f \) is a permutation of \( {\mathbf{N}}^{ + } \) (that is, a one-one mapping of \( {\mathbf{N}}^{ + } \) onto itself). A theorem first proved by Riemann shows that if \( \sum {a}_{n} \) is a convergent, but not absolutely convergent, series of real numbers, then for each real number \( s \) there exists a rearrangement of \( \sum {a}_{n} \) that converges to \( s \) . The second exercise in the next set leads you through a proof of this remarkable result. ## (1.2.17) Exercises . 1 Prove that if \( \sum {a}_{n} \) is absolutely convergent, with sum \( s \), then any rearrangement of \( \sum {a}_{n} \) converges to \( s \) . (Given a permutation \( f \) of \( {\mathbf{N}}^{ + } \) and a positive number \( \varepsilon \), choose \( N \) such that \( \mathop{\sum }\limits_{{n = N + 1}}^{\infty }\left| {a}_{n}\right| < \varepsilon \) . Then choose \( M \geq N \) such that \( \{ 1,2,\ldots, N\} \subset \{ f\left( 1\right), f\left( 2\right) ,\ldots, f\left( M\right) \} \) . Show that \( \left| {\mathop{\sum }\limits_{{n = 1}}^{m}{a}_{f\left( n\right) } - s}\right| < {2\varepsilon } \) for all \( m \geq M \) .) .2 Let \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n} \) be an infinite series of real numbers that converges but is not absolutely convergent. For each \( n \) define \( {a}_{n}^{ + } = \max \left\{ {{a}_{n},0}\right\} ,{a}_{n}^{ - } = \) \( \min \left\{ {{a}_{n},0}\right\} \) . Prove that the partial sums of the series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}^{ + } \) and \( \mathop{\sum }\limits_{{n = 1}}^{\in

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\( s \), then any rearrangement of \( \sum {a}_{n} \) converges to \( s \) . (Given a permutation \( f \) of \( {\mathbf{N}}^{ + } \) and a positive number \( \varepsilon \), choose \( N \) such that \( \mathop{\sum }\limits_{{n = N + 1}}^{\infty }\left| {a}_{n}\right| < \varepsilon \) . Then choose \( M \geq N \) such that \( \{ 1,2,\ldots, N\} \subset \{ f\left( 1\right), f\left( 2\right) ,\ldots, f\left( M\right) \} \) . Show that \( \left| {\mathop{\sum }\limits_{{n = 1}}^{m}{a}_{f\left( n\right) } - s}\right| < {2\varepsilon } \) for all \( m \geq M \) .) .2 Let \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n} \) be an infinite series of real numbers that converges but is not absolutely convergent. For each \( n \) define \( {a}_{n}^{ + } = \max \left\{ {{a}_{n},0}\right\} ,{a}_{n}^{ - } = \) \( \min \left\{ {{a}_{n},0}\right\} \) . Prove that the partial sums of the series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}^{ + } \) and \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}^{ - } \) form increasing unbounded sequences. Now let \( s \) be any real number. Let \( {n}_{0} \) and \( {m}_{0} \) both equal 0, and let \( {n}_{1} \) be the least positive integer such that \[ {a}_{1}^{ + } + {a}_{2}^{ + } + \cdots + {a}_{{n}_{1}}^{ + } > s. \] Show how to construct strictly increasing sequences \( {\left( {n}_{k}\right) }_{k = 0}^{\infty } \) and \( {\left( {m}_{k}\right) }_{k = 0}^{\infty } \) of positive integers such that for each \( N \geq 0 \) , \[ \mathop{\sum }\limits_{{k = 0}}^{{N - 1}}\left( {\left( {{a}_{{n}_{k} + 1}^{ + } + \cdots + {a}_{{n}_{k + 1}}^{ + }}\right) + \left( {{a}_{{m}_{k} + 1}^{ - } + \cdots + {a}_{{m}_{k + 1}}^{ - }}\right) }\right) < s \] and \[ \mathop{\sum }\limits_{{k = 0}}^{{N - 1}}\left( {\left( {{a}_{{n}_{k} + 1}^{ + } + \cdots + {a}_{{n}_{k + 1}}^{ + }}\right) + \left( {{a}_{{m}_{k} + 1}^{ - } + \cdots + {a}_{{m}_{k + 1}}^{ - }}\right) }\right) \] \[ + \left( {{a}_{{n}_{N} + 1}^{ + } + \cdots + {a}_{{n}_{N + 1}}^{ + }}\right) > s. \] Hence obtain a rearrangement of \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n} \) that converges to \( s \) . .3 Let \( s \) be the sum of the series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{\left( -1\right) }^{n - 1}/n \) . Show that the series \[ 1 - \frac{1}{2} - \frac{1}{4} + \frac{1}{3} - \frac{1}{6} - \frac{1}{8} + \frac{1}{5} - \frac{1}{10} - \frac{1}{12} + \cdots \] converges to \( \frac{1}{2}\log 2 \) . ## 1.3 Open and Closed Subsets of the Line In this section we introduce the fundamental topological notions of "open set" and "closed set" in \( \mathbf{R} \), notions that readily generalise in later, more abstract contexts. A subset \( A \) of \( \mathbf{R} \) is said to be open (in \( \mathbf{R} \) ) if to each \( x \in A \) there corresponds \( r > 0 \) such that the open interval \( \left( {x - r, x + r}\right) \) is contained in \( A \) -or, equivalently, such that \( y \in A \) whenever \( \left| {x - y}\right| < r \) . ## (1.3.1) Exercises . 1 Prove that \( \mathbf{R} \) itself, the empty set \( \varnothing \), and all open intervals are open subsets of \( \mathbf{R} \) . .2 Give an example of a sequence of open subsets of \( \mathbf{R} \) whose intersection is not open. The first result in this section describes the two fundamental properties of open sets. (1.3.2) Proposition. The union of any family of open sets is open. The intersection of any finite family of open sets is open. Proof. Let \( {\left( {A}_{i}\right) }_{i \in I} \) be any family of open sets. If \( x \) belongs to the union \( U \) of this family, then \( x \in {A}_{i} \) for some \( i \) . As \( {A}_{i} \) is open, there exists \( r > 0 \) such that \[ \left( {x - r, x + r}\right) \subset {A}_{i} \subset U. \] Hence \( U \) is open. Now let \( {A}_{1},\ldots ,{A}_{n} \) be finitely many open sets, and consider any \( x \) in their intersection. For each \( i \), since \( x \in {A}_{i} \) and \( {A}_{i} \) is open, there exists \( {r}_{i} > 0 \) such that \( y \in {A}_{i} \) whenever \( \left| {x - y}\right| < {r}_{i} \) . Let \[ r = \min \left\{ {{r}_{1},\ldots ,{r}_{n}}\right\} > 0. \] If \( \left| {x - y}\right| < r \), then \( y \in {A}_{i} \) for each \( i \), so \( y \in \mathop{\bigcap }\limits_{{i = 1}}^{n}{A}_{i} \) . Hence \( \mathop{\bigcap }\limits_{{i = 1}}^{n}{A}_{i} \) is open. In view of Exercise (1.3.1: 2), we cannot drop the word "finite" from the hypothesis of the second part of Proposition (1.3.2). Our next aim is to characterise open sets in \( \mathbf{R} \) ; to achieve this, we first characterise intervals. A nonempty subset \( S \) of \( \mathbf{R} \) is said to have the intermediate value property if \( \left( {a, b}\right) \subset S \) whenever \( a \in S, b \in S \), and \( a < b \) . Of course, as we show in Section 4, this notion is connected with the Intermediate Value Theorem of elementary calculus. (1.3.3) Proposition. A subset \( S \) of \( \mathbf{R} \) has the intermediate value property if and only if it is an interval. Proof. It is clear that every interval in \( \mathbf{R} \) has the intermediate value property. Conversely, suppose that \( S \subset \mathbf{R} \) has that property. Assume, to begin with, that \( S \) is bounded, and let \( a \) be its infimum and \( b \) its supremum. Note that \( x \notin S \) if either \( x < a \) or \( x > b \) . If \( a \) and \( b \) both belong to \( S \), then by the intermediate value property, so does every point of \( \left\lbrack {a, b}\right\rbrack \) ; whence \( S = \left\lbrack {a, b}\right\rbrack \) . If \( a \in S \) and \( b \notin S \), consider any \( x \) such that \( a \leq x < b \) . By the definition of "supremum", there exists \( s \in S \) such that \( a \leq x < s \) ; the intermediate value property now ensures that \( x \in S \) ; whence \( S = \lbrack a, b) \) . Similarly, if \( a \notin S \) and \( b \in S \), then \( S = (a, b\rbrack \) . The remaining cases are left as exercises. ## (1.3.4) Exercises . 1 Prove that a nonempty open subset of \( \mathbf{R} \) with the intermediate value property is an open interval. .2 Complete the proof of Proposition (1.3.3) in the remaining cases. .3 Let \( I, J \) be open intervals with nonempty intersection. Prove that \( I \cup J \) and \( I \cap J \) are open intervals. \( \textbf{(1.3.5) Lemma. }\;\textit{A nonempty family of pairwise-disjoint open intervals} \) of \( \mathbf{R} \) is countable. Proof. Let \( \mathcal{F} \) be a nonempty family of pairwise-disjoint open intervals in \( \mathbf{R} \), and note that, by Exercise (1.1.1:19), each of these intervals contains a rational number. The Axiom of Choice (see Appendix B) ensures that there is a function \( f : \mathcal{F} \rightarrow \mathbf{Q} \) such that \( f\left( I\right) \in I \) for each \( I \in \mathcal{F} \) . Since the sets in \( \mathcal{F} \) are pairwise disjoint, \( f \) is one-one and so has an inverse function \( g \) mapping \( f\left( \mathcal{F}\right) \) onto \( \mathcal{F} \) . As \( \mathbf{Q} \) is countable and \( f\left( \mathcal{F}\right) \subset \mathbf{Q} \), there exists a mapping \( h \) of \( {\mathbf{N}}^{ + } \) onto \( f\left( \mathcal{F}\right) \) ; the composite function \( g \circ h \) then maps \( {\mathbf{N}}^{ + } \) onto \( \mathcal{F} \), which is therefore countable. (1.3.6) Proposition. A nonempty subset of \( \mathbf{R} \) is open if and only if it is the union of a sequence of pairwise-disjoint open intervals. Proof. It follows from Proposition (1.3.2) and Exercise (1.3.1:1) that the union of any family of open intervals is an open set. Conversely, given a nonempty open subset \( S \) of \( \mathbf{R} \), define a binary relation \( \sim \) on \( S \) by setting \( x \sim y \) if and only if there exists an open interval \( I \subset S \) such that \( x, y \in \) \( I \) . Then \( \sim \) is an equivalence relation: it is straightforward to prove the reflexivity and symmetry of \( \sim \), and its transitivity follows from Exercise (1.3.4:3). Clearly, the equivalence class \( \dot{x} \) of \( x \) under \( \sim \) is a union of open intervals and is therefore an open set. Consider points \( y, z \in \dot{x} \) and a real number \( t \) with \( y < t < z \) . Choosing open intervals \( {I}_{y},{I}_{z} \subset S \) such that \( x, y \in {I}_{y} \) and \( x, z \in {I}_{z} \), we see from Exercise (1.3.4: 3) that \( {I}_{y} \cup {I}_{z} \) is an open interval; so \( t \in \left( {y, z}\right) \subset {I}_{y} \cup {I}_{z} \), and therefore either \( x, t \in {I}_{y} \subset S \) or else \( x, t \in {I}_{z} \subset S \) . Hence \( \dot{x} \) has the intermediate value property. It follows from Exercise (1.3.4:1) that \( \dot{x} \) is an open interval. Since any two distinct equivalence classes under \( \sim \) are disjoint, we now see that \[ S = \mathop{\bigcup }\limits_{{x \in S}}\dot{x} \] is a union of pairwise-disjoint open intervals. Reference to Lemma (1.3.5) completes the proof. A real number \( x \) is an interior point of a set \( S \subset \mathbf{R} \) if there exists \( r > 0 \) such that \( \left( {x - r, x + r}\right) \subset S \) . The set of all interior points of \( S \) is called the interior of \( S \), and is written \( {S}^{ \circ } \) . By a neighbourhood of \( x \) we mean a set containing \( x \) in its interior. ## (1.3.7) Exercises . 1 Let \( S \) be a nonempty open subset of \( \mathbf{R} \), and for each \( x \in S \) consider the sets \[ {U}_{x} = \{ t \in \mathbf{R} : \left( {x, t}\right) \subset S\} \] \[ {L}_{x} = \{ s \in \mathbf{R} : \left( {s, x}\right) \subset S\} . \] Let \( a = \inf {L}_{x}, b = \sup {U}_{x} \), and \( {I}_{x} = \left( {a, b}\right) \), where \( a = - \infty \) if \( {L}_{x} \) is not bounded below, and \( b = \infty \) if \( {U}_{x} \) is not bounded above. Give another proof of Proposition (1.3.6) by showing that \( {\left( {I}_{x}\right) }_{x \in S} \) is a family of disjoint open intervals whose un

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oof. A real number \( x \) is an interior point of a set \( S \subset \mathbf{R} \) if there exists \( r > 0 \) such that \( \left( {x - r, x + r}\right) \subset S \) . The set of all interior points of \( S \) is called the interior of \( S \), and is written \( {S}^{ \circ } \) . By a neighbourhood of \( x \) we mean a set containing \( x \) in its interior. ## (1.3.7) Exercises . 1 Let \( S \) be a nonempty open subset of \( \mathbf{R} \), and for each \( x \in S \) consider the sets \[ {U}_{x} = \{ t \in \mathbf{R} : \left( {x, t}\right) \subset S\} \] \[ {L}_{x} = \{ s \in \mathbf{R} : \left( {s, x}\right) \subset S\} . \] Let \( a = \inf {L}_{x}, b = \sup {U}_{x} \), and \( {I}_{x} = \left( {a, b}\right) \), where \( a = - \infty \) if \( {L}_{x} \) is not bounded below, and \( b = \infty \) if \( {U}_{x} \) is not bounded above. Give another proof of Proposition (1.3.6) by showing that \( {\left( {I}_{x}\right) }_{x \in S} \) is a family of disjoint open intervals whose union is \( S \) . 1. Analysis on the Real Line .2 Prove that the interior of an open, closed, or half open interval with endpoints \( a \) and \( b \), where \( a < b \), is the open interval \( \left( {a, b}\right) \) . .3 Show that \( {\left( {S}^{ \circ }\right) }^{ \circ } = {S}^{ \circ } \) . .4 Prove that \( {S}^{ \circ } \) is the largest open set contained in \( S \) -in other words, that (i) \( {S}^{ \circ } \) is open and \( {S}^{ \circ } \subset S \) ; (ii) if \( A \) is open and \( A \subset S \), then \( A \subset {S}^{ \circ } \) . .5 Prove that \( S \) is open if and only if \( S \subset {S}^{ \circ } \) . .6 Prove that \( {S}^{ \circ } \) is the union of the open sets contained in \( S \) . .7 Prove that (i) if \( S \subset T \), then \( {S}^{ \circ } \subset {T}^{ \circ } \) ; (ii) \( {\left( S \cap T\right) }^{ \circ } = {S}^{ \circ } \cap {T}^{ \circ } \) . .8 Prove that \( U \) is a neighbourhood of \( x \in \mathbf{R} \) if and only if there is an open set \( A \) such that \( x \in A \subset U \) . Let \( x \) be a real number, and \( S \) a subset of \( \mathbf{R} \) . We call \( x \) a cluster point of \( S \) if each neighbourhood of \( x \) has a nonempty intersection with \( S \) ; or, equivalently, if for each \( \varepsilon > 0 \) there exists \( y \in S \) such that \( \left| {x - y}\right| < \varepsilon \) . The closure of \( S \) (in \( \mathbf{R} \) ) is the set of all cluster points of \( S \), and is denoted by \( \bar{S} \) or \( {S}^{ - }.S \) is said to be closed if \( S = \bar{S} \) . ## (1.3.8) Exercises .1 Is \( \mathbf{Q} \) closed in \( \mathbf{R} \) ? Is it open in \( \mathbf{R} \) ? .2 Show that the closure of any interval with endpoints \( a \) and \( b \), where \( a < b \), is the closed interval \( \left\lbrack {a, b}\right\rbrack \) . .3 Show that \( \overline{\left( \bar{S}\right) } = \bar{S} \) . .4 Prove that \( \bar{S} \) is the smallest closed set containing \( S \) -in other words, that (i) \( \bar{S} \) is closed and \( S \subset \bar{S} \) ; (ii) if \( A \) is closed and \( S \subset A \), then \( \bar{S} \subset A \) . .5 Prove that \( S \) is closed if and only if \( \bar{S} \subset S \) . .6 Prove that \( \bar{S} \) is the intersection of the closed sets containing \( S \) . .7 Prove the following. (i) If \( S \subset T \), then \( \bar{S} \subset \bar{T} \) ; (ii) \( \overline{S \cup T} = \bar{S} \cup \bar{T} \) . .8 Prove that (i) the complement of \( {S}^{ \circ } \) is the closure of \( \mathbf{R} \smallsetminus S \) ; (ii) the complement of \( \bar{S} \) is the interior of \( \mathbf{R} \smallsetminus S \) . .9 The boundary, or frontier, of a set \( S \subset \mathbf{R} \) is the intersection of the closures of \( S \) and \( \mathbf{R} \smallsetminus S \) . Describe the boundary of each of the following sets: \( \mathbf{R},\varnothing ,(a, b\rbrack \) (where \( a < b \) ), \( \mathbf{Q} \) . .10 Prove that \( a \) belongs to the boundary of \( S \smallsetminus \{ a\} \) if and only if \( a \in \) \( \overline{S\smallsetminus \{ a\} } \) . .11 Let \( C \) be the Cantor set-that is, the subset of \( \left\lbrack {0,1}\right\rbrack \) consisting of all numbers that have a ternary (base 3) expansion \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}{3}^{-n} \) with \( {a}_{n} \in \{ 0,2\} \) for each \( n \) . Prove that (i) if \( a, b \) are two numbers in \( C \) that differ in their \( m \) th ternary places, then \( \left| {a - b}\right| \geq {3}^{-m} \) ; (ii) \( C \) is a closed subset of \( \mathbf{R} \) ; (iii) \( C \) has an empty interior. What is the boundary of \( C \) ? ## (1.3.9) Proposition. \( S \) is closed if and only if \( \mathbf{R} \smallsetminus S \) is open. Proof. Suppose that \( S \) is closed, and consider any \( x \in \mathbf{R} \smallsetminus S \) . Since \( S = \bar{S} \) , \( x \) is not a cluster point of \( S \) ; so there exists a neighbourhood \( U \) of \( x \) that is disjoint from \( S \) . By Exercise (1.3.7:8), there is an open set \( A \) such that \( x \in A \subset U \) . Then \( A \cap S = \varnothing \), so \( A \subset \mathbf{R} \smallsetminus S \) ; whence, by Exercise (1.3.7: 8), \( \mathbf{R} \smallsetminus S \) is a neighbourhood of \( x \), and therefore \( x \in {\left( \mathbf{R} \smallsetminus S\right) }^{ \circ } \) . Since \( x \) is any element of \( \mathbf{R} \smallsetminus S \), we conclude that \( \mathbf{R} \smallsetminus S \) is open. Conversely, suppose that \( \mathbf{R} \smallsetminus S \) is open. Then by Exercise (1.3.7:8), \( \mathbf{R} \smallsetminus S \) is a neighbourhood of each of its points. Since \( \mathbf{R} \smallsetminus S \) is disjoint from \( S \), it follows that no point of \( \mathbf{R} \smallsetminus S \) is in the closure of \( S \) . Thus if \( x \in \bar{S} \), then \( x \notin \mathbf{R} \smallsetminus S \) and so \( x \in S \) . Hence \( \bar{S} \subset S \), and therefore, by Exercise (1.3.8: 5), \( S \) is closed. (1.3.10) Proposition. The intersection of a family of closed sets is closed. The union of a finite family of closed sets is closed. Proof. Let \( {\left( {C}_{i}\right) }_{i \in I} \) be any family of closed sets, and for each \( i \) let \( {A}_{i} \) be the complement of \( {C}_{i} \) . Then \[ \mathop{\bigcap }\limits_{{i \in I}}{C}_{i} = \mathbf{R} \smallsetminus \left( {\mathop{\bigcup }\limits_{{i \in I}}{A}_{i}}\right) \] Since, by Proposition (1.3.9), each \( {A}_{i} \) is open, Proposition (1.3.2) shows that \( \mathop{\bigcup }\limits_{{i \in I}}{A}_{i} \) is open; whence, again by Proposition (1.3.9), its complement is closed. This completes the first part of the proof; the second is left as an exercise. ## (1.3.11) Exercises .1 Complete the proof of Proposition (1.3.10). .2 Give an example of a sequence of closed sets whose union is not closed. Which subsets of \( \mathbf{R} \) are both open and closed? Before answering this question, we prove a simple lemma. (1.3.12) Lemma. If \( T \) is a nonempty open subset of \( \mathbf{R} \) that is bounded above (respectively, below), then \( \sup T \notin T \) (respectively, \( \inf T \notin T \) ). Proof. Consider, for example, the case where \( T \) is bounded above. Suppose that \( M = \sup T \) belongs to \( T \) . Since \( T \) is open, \( \left( {M - r, M + r}\right) \subset T \) for some \( r > 0 \) . Hence \( M + \frac{1}{2}r \in T \), which is absurd as \( M + \frac{1}{2}r > \sup T \) . Hence, in fact, \( M \notin T \) . (1.3.13) Proposition. R and \( \varnothing \) are the only subsets of \( \mathbf{R} \) that are both open and closed in \( \mathbf{R} \) . Proof. Exercise (1.3.1: 1) and Proposition (1.3.9) show that \( \mathbf{R} \) and \( \varnothing \) are both open and closed in \( \mathbf{R} \) . Let \( S \) be a nonempty set that is both open and closed, and note that, by Proposition (1.3.9), \( \mathbf{R} \smallsetminus S \) is also both open and closed. Suppose \( \mathbf{R} \smallsetminus S \) is nonempty. Choosing \( a \in S \) and \( b \in \mathbf{R} \smallsetminus S \), we have either \( a < b \) or \( a > b \) . Without loss of generality we take the former case, so that \[ T = \{ x \in \mathbf{R} \smallsetminus S : x > a\} \] is nonempty and bounded below. By Proposition (1.3.2), \( T \) is also open, being the intersection of the open sets \( \left( {a,\infty }\right) \) and \( \mathbf{R} \smallsetminus S \) . Let \( m = \inf T \) . Since \( S \) is open, there exists \( r > 0 \) such that \( \left( {a - r, a + r}\right) \subset S \) ; whence \( m \geq a + r > a \) . Since, by Lemma (1.3.12), \( m \notin T \), it follows that \( m \notin \mathbf{R} \smallsetminus S \) and therefore that \( m \in S \) . But \( S \) is open, so there exists \( \varepsilon > 0 \) such that \( \left( {m - \varepsilon, m + \varepsilon }\right) \subset S \) ; this is impossible, since the definition of "infimum" ensures that there exists \( t \in \mathbf{R} \smallsetminus S \) such that \( t < m + \varepsilon \) . This contradiction shows that \( \mathbf{R} \smallsetminus S \) is empty; whence \( S = \mathbf{R} \) . ## 1.4 Limits and Continuity Let \( I \) be an interval in \( \mathbf{R}, a \) a point of the closure of \( I \), and \( f \) a real-valued function whose domain includes \( I \) but not necessarily \( a \) . A real number \( l \) is called the limit of \( f\left( x\right) \) as \( x \) tends to a in \( I \), or the limit of \( f \) at a (relative to \( I \) ), if to each \( \varepsilon > 0 \) there corresponds \( \delta > 0 \) such that \( \left| {f\left( x\right) - l}\right| < \varepsilon \) whenever \( x \in I \) and \( 0 < \left| {x - a}\right| < \delta \) . We then write \[ f\left( x\right) \rightarrow l\text{as}x \rightarrow a, x \in I \] or \[ \mathop{\lim }\limits_{{x \rightarrow a, x \in I}}f\left( x\right) = l \] and we say that \( f\left( x\right) \) tends to \( l \) as \( x \) tends to a through values in \( I \) . The following

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ch that \( t < m + \varepsilon \) . This contradiction shows that \( \mathbf{R} \smallsetminus S \) is empty; whence \( S = \mathbf{R} \) . ## 1.4 Limits and Continuity Let \( I \) be an interval in \( \mathbf{R}, a \) a point of the closure of \( I \), and \( f \) a real-valued function whose domain includes \( I \) but not necessarily \( a \) . A real number \( l \) is called the limit of \( f\left( x\right) \) as \( x \) tends to a in \( I \), or the limit of \( f \) at a (relative to \( I \) ), if to each \( \varepsilon > 0 \) there corresponds \( \delta > 0 \) such that \( \left| {f\left( x\right) - l}\right| < \varepsilon \) whenever \( x \in I \) and \( 0 < \left| {x - a}\right| < \delta \) . We then write \[ f\left( x\right) \rightarrow l\text{as}x \rightarrow a, x \in I \] or \[ \mathop{\lim }\limits_{{x \rightarrow a, x \in I}}f\left( x\right) = l \] and we say that \( f\left( x\right) \) tends to \( l \) as \( x \) tends to a through values in \( I \) . The following are the most important cases of this definition. - \( a \in {I}^{ \circ } \) : in this case we use the simpler notations \[ f\left( x\right) \rightarrow l\text{ as }x \rightarrow a \] and \[ \mathop{\lim }\limits_{{x \rightarrow a}}f\left( x\right) = l \] - \( I = \left( {c, a}\right) \) for some \( c < a \) (where \( c \) could be \( - \infty \) ): in this case we call \( l \) the left-hand limit of \( f \) as \( x \) tends to \( a \) ; we say that \( f\left( x\right) \) tends to \( l \) as \( x \) tends to a from the left (or from below); and we use the notations \[ f\left( x\right) \rightarrow l\text{ as }x \rightarrow {a}^{ - } \] and \[ f\left( {a}^{ - }\right) = \mathop{\lim }\limits_{{x \rightarrow {a}^{ - }}}f\left( x\right) = l. \] - \( I = \left( {a, b}\right) \) for some \( b > a \) (where \( b \) could be \( \infty \) ): in this case we call \( l \) the right-hand limit of \( f \) as \( x \) tends to \( a \) ; we say that \( f\left( x\right) \) tends to \( l \) as \( x \) tends to a from the right (or from above); and we use the notations \[ f\left( x\right) \rightarrow l\text{ as }x \rightarrow {a}^{ + } \] and \[ f\left( {a}^{ + }\right) = \mathop{\lim }\limits_{{x \rightarrow {a}^{ + }}}f\left( x\right) = l. \] We stress that although, in our definition of "limit", \( f\left( x\right) \) is defined for all \( x \) in \( I \) that are distinct from but sufficiently close to \( a, f\left( a\right) \) need not be defined. For example, in elementary calculus courses we learn that \[ \mathop{\lim }\limits_{{x \rightarrow 0}}\frac{\sin x}{x} = 1 \] even though \( \left( {\sin x}\right) /x \) is not defined at \( x = 0 \) . (1.4.1) Proposition. If \( \mathop{\lim }\limits_{{x \rightarrow a, x \in I}}f\left( x\right) = l \) and \( \mathop{\lim }\limits_{{x \rightarrow a, x \in I}}f\left( x\right) = {l}^{\prime } \) , then \( l = {l}^{\prime } \) . Proof. Given \( \varepsilon > 0 \), choose \( {\delta }_{f},{\delta }_{g} > 0 \) such that - if \( x \in I \) and \( 0 < \left| {x - a}\right| < {\delta }_{f} \), then \( \left| {f\left( x\right) - l}\right| < \varepsilon /2 \), and - if \( x \in I \) and \( 0 < \left| {x - a}\right| < {\delta }_{g} \), then \( \left| {f\left( x\right) - {l}^{\prime }}\right| < \varepsilon /2 \) . Setting \( \delta = \min \left\{ {{\delta }_{f},{\delta }_{g}}\right\} \), consider any \( x \in I \) such that \( 0 < \left| {x - a}\right| < \delta \) . We have \[ \left| {l - {l}^{\prime }}\right| \leq \left| {f\left( x\right) - l}\right| + \left| {f\left( x\right) - {l}^{\prime }}\right| < \frac{\varepsilon }{2} + \frac{\varepsilon }{2} = \varepsilon . \] Since \( \varepsilon > 0 \) is arbitrary, it follows from Exercise (1.1.2:4) that \( l = {l}^{\prime } \) . (1.4.2) Proposition. If \( \mathop{\lim }\limits_{{x \rightarrow a, x \in I}}f\left( x\right) = l \) and \( \mathop{\lim }\limits_{{x \rightarrow a, x \in I}}g\left( x\right) = m \) , then as \( x \rightarrow a \) through values in \( I \) , \[ f\left( x\right) + g\left( x\right) \rightarrow l + m \] \[ f\left( x\right) - g\left( x\right) \rightarrow l - m \] \[ f\left( x\right) g\left( x\right) \rightarrow {lm} \] \[ \max \{ f\left( x\right), g\left( x\right) \} \rightarrow \max \{ l, m\} \] \[ \min \{ f\left( x\right), g\left( x\right) \} \rightarrow \min \{ l, m\} \] \[ \left| {f\left( x\right) }\right| \rightarrow \left| l\right| \] If also \( m \neq 0 \), then \[ \mathop{\lim }\limits_{{x \rightarrow a, x \in I}}\frac{f\left( x\right) }{g\left( x\right) } = \frac{l}{m} \] Proof. See Exercise (1.4.3: 3). ## (1.4.3) Exercises . 1 Define precisely what it means to say that \( f\left( x\right) \) does not converge to any limit as \( x \) tends to a through values in \( I \) . In other words, give the formal negation of the definition of "convergent". .2 Let the function \( f \) be defined in an interval whose interior contains \( a \) . Prove that \( \mathop{\lim }\limits_{{x \rightarrow a}}f\left( x\right) = l \) if and only if \( \mathop{\lim }\limits_{{x \rightarrow {a}^{ + }}}f\left( x\right) \) and \( \mathop{\lim }\limits_{{x \rightarrow {a}^{ - }}}f\left( x\right) \) exist and equal \( l \) . .3 Prove Proposition (1.4.2). .4 Use the definition of "limit" to prove that \( \mathop{\lim }\limits_{{x \rightarrow a, x \in \mathbf{R}}}p\left( x\right) = p\left( a\right) \) for any polynomial function \( p \) and any \( a \in \mathbf{R} \) . .5 Let \( p, q \) be polynomial functions, and \( a \) a real number such that \( q\left( a\right) \neq \) 0 . Prove that \[ \mathop{\lim }\limits_{{x \rightarrow a, x \in \mathbf{R}}}\frac{p\left( x\right) }{q\left( x\right) } = \frac{p\left( a\right) }{q\left( a\right) }. \] .6 Prove that \( \mathop{\lim }\limits_{{x \rightarrow a, x \in I}}f\left( x\right) = l \) if and only if for each sequence \( \left( {a}_{n}\right) \) of elements of \( I \) that converges to \( a \), the sequence \( \left( {f\left( {a}_{n}\right) }\right) \) converges to \( l \) . (For "only if", use a proof by contradiction.) .7 Suppose that \( \mathop{\lim }\limits_{{x \rightarrow a, x \in I}}f\left( x\right) > r \) . Prove that there exists \( \delta > 0 \) such that if \( x \in I \) and \( 0 < \left| {x - a}\right| < \delta \), then \( f\left( x\right) > r \) . .8 Let \( f \) be a real-valued function whose domain includes an interval of the form \( \left( {s,\infty }\right) \), and let \( l \in \mathbf{R} \) . We say that \( f\left( x\right) \) tends to \( l \) as \( x \) tends to \( \infty \) if to each \( \varepsilon > 0 \) there corresponds \( K > 0 \) such that \( \left| {f\left( x\right) - l}\right| < \varepsilon \) whenever \( x > K \) ; we then write \[ f\left( x\right) \rightarrow l\text{as}x \rightarrow \infty \] or \[ \mathop{\lim }\limits_{{x \rightarrow \infty }}f\left( x\right) = l \] Convince yourself that analogues of Propositions (1.4.1) and (1.4.2) hold for limits as \( x \) tends to \( \infty \) . Define the notion \( f\left( x\right) \) tends to \( l \) as \( x \) tends to \( - \infty \), written \[ f\left( x\right) \rightarrow l\text{as}x \rightarrow - \infty \] or \[ \mathop{\lim }\limits_{{x \rightarrow - \infty }}f\left( x\right) = l \] and convince yourself that analogues of Propositions (1.4.1) and (1.4.2) hold for this notion also. .9 Formulate definitions of the following notions, where \( I \) is an interval. (i) \( f\left( x\right) \rightarrow \infty \) as \( x \rightarrow a \) through values in \( I \) . (ii) \( f\left( x\right) \rightarrow - \infty \) as \( x \rightarrow a \) through values in \( I \) . (iii) \( f\left( x\right) \rightarrow \infty \) as \( x \rightarrow \infty \) . (iv) \( f\left( x\right) \rightarrow \infty \) as \( x \rightarrow - \infty \) . (v) \( f\left( x\right) \rightarrow - \infty \) as \( x \rightarrow \infty \) . (vi) \( f\left( x\right) \rightarrow - \infty \) as \( x \rightarrow - \infty \) . .10 Prove that if \( a > 1 \), then \( {a}^{x} \rightarrow 0 \) as \( x \rightarrow - \infty \), and \( {a}^{x} \rightarrow \infty \) as \( x \rightarrow \infty \) . What happens to \( {a}^{x} \) as \( x \rightarrow \pm \infty \) when \( 0 < a < 1 \) ? (Note Exercise (1.2.3:3).) .11 Prove that if \( a > 1 \), then \( {\log }_{a}x \rightarrow 0 \) as \( x \rightarrow - \infty \), and \( {\log }_{a}x \rightarrow \infty \) as \( x \rightarrow \infty \) . What happens to \( {\log }_{a}x \) as \( x \rightarrow \pm \infty \) if \( 0 < a < 1 \) ? .12 Let \( f \) be a real-valued function, and, where appropriate, define \[ \mathop{\lim }\limits_{{x \rightarrow \xi }}\sup f\left( x\right) = \mathop{\inf }\limits_{{r > 0}}\sup \{ f\left( x\right) : 0 < \left| {x - \xi }\right| < r\} \] \[ \begin{matrix} \mathop{\lim }\limits_{{x \rightarrow {\xi }^{ + }}}\sup f\left( x\right) & = & \mathop{\inf }\limits_{{r > 0}}\sup \left\{ {f\left( x\right) : 0 < x - \xi < r}\right\} , \end{matrix} \] \[ \begin{matrix} \mathop{\lim }\limits_{{x \rightarrow {\xi }^{ - }}}\sup f\left( x\right) & = & \mathop{\inf }\limits_{{r > 0}}\sup \left\{ {f\left( x\right) : 0 < \xi - x < r}\right\} . \end{matrix} \] For example, in order that \( \mathop{\lim }\limits_{{x \rightarrow {\xi }^{ - }}}\sup f\left( x\right) \) be defined, it is necessary that \( f \) be defined and bounded \( {}^{4} \) on some interval of the form \( \left( {\xi - r,\xi }\right) \) , where \( r > 0 \) . Prove the following. (a) \( \mathop{\lim }\limits_{{x \rightarrow \xi }}\sup f\left( x\right) \leq M \) if and only if for each \( \varepsilon > 0 \) there exists \( \delta > 0 \) such that \( f\left( x\right) < M + \varepsilon \) whenever \( 0 < \left| {x - \xi }\right| < \delta \) . (b) \( \mathop{\lim }\limits_{{x \rightarrow \xi }}\sup f\left( x\right) \geq M \) if and only if for each pair of positive numbers \( \varepsilon ,\delta \) there exists \( x \) such that \( 0 < \left| {x - \xi }\right| < \delta \) and \( f\left( x\right) > \) \( M - \varepsilon \) . Formu

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{\xi }^{ - }}}\sup f\left( x\right) & = & \mathop{\inf }\limits_{{r > 0}}\sup \left\{ {f\left( x\right) : 0 < \xi - x < r}\right\} . \end{matrix} \] For example, in order that \( \mathop{\lim }\limits_{{x \rightarrow {\xi }^{ - }}}\sup f\left( x\right) \) be defined, it is necessary that \( f \) be defined and bounded \( {}^{4} \) on some interval of the form \( \left( {\xi - r,\xi }\right) \) , where \( r > 0 \) . Prove the following. (a) \( \mathop{\lim }\limits_{{x \rightarrow \xi }}\sup f\left( x\right) \leq M \) if and only if for each \( \varepsilon > 0 \) there exists \( \delta > 0 \) such that \( f\left( x\right) < M + \varepsilon \) whenever \( 0 < \left| {x - \xi }\right| < \delta \) . (b) \( \mathop{\lim }\limits_{{x \rightarrow \xi }}\sup f\left( x\right) \geq M \) if and only if for each pair of positive numbers \( \varepsilon ,\delta \) there exists \( x \) such that \( 0 < \left| {x - \xi }\right| < \delta \) and \( f\left( x\right) > \) \( M - \varepsilon \) . Formulate appropriate definitions of the quantities \( \mathop{\lim }\limits_{{x \rightarrow \xi }}\inf f\left( x\right) \) , \( \mathop{\lim }\limits_{{x \rightarrow {\xi }^{ + }}}\inf f\left( x\right) \), and \( \mathop{\lim }\limits_{{x \rightarrow {\xi }^{ - }}}\inf f\left( x\right) \) . Prove that (c) \( \mathop{\lim }\limits_{{x \rightarrow \xi }}\inf f\left( x\right) \leq \mathop{\lim }\limits_{{x \rightarrow \xi }}\sup f\left( x\right) \), and these two numbers are equal if and only if \( l = \mathop{\lim }\limits_{{x \rightarrow \xi }}f\left( x\right) \) exists, in which case the numbers equal \( l \) . An important special case of the notion of a limit occurs when the function \( f \) is defined at the point \( a \) that we are approaching. A function \( f \) defined in some neighbourhood of \( a \) is said to be continuous at \( a \) if \( f\left( x\right) \rightarrow f\left( a\right) \) as \( x \rightarrow a \) ; in other words, if for each \( \varepsilon > 0 \) there exists \( \delta > 0 \) such that \( \left| {f\left( x\right) - f\left( a\right) }\right| < \varepsilon \) whenever \( \left| {x - a}\right| < \delta \) . We also say that \( f \) is - continuous on the left at \( a \) if \( f \) is defined on the interval \( (a - r, a\rbrack \) for some \( r > 0 \) and \[ f\left( {a}^{ - }\right) = \mathop{\lim }\limits_{{x \rightarrow {a}^{ - }}}f\left( x\right) = f\left( a\right) \] - continuous on the right at \( a \) if \( f \) is defined on the interval \( \lbrack a, a + r) \) for some \( r > 0 \) and \[ f\left( {a}^{ + }\right) = \mathop{\lim }\limits_{{x \rightarrow {a}^{ + }}}f\left( x\right) = f\left( a\right) \] \( {}^{4} \) By allowing the lim sup quantities to take the values \( \pm \infty \), in a sense that is made precise in Section 3.1, we can remove the restriction that \( f \) be bounded near \( \xi \) . - continuous on the interval \( I \) if \( \mathop{\lim }\limits_{{x \rightarrow t, x \in I}}f\left( x\right) = f\left( t\right) \) for each \( t \in I \) . Note that the last definition takes care of the one-sided continuity of \( f \) at those endpoints of \( I \), if any, that belong to \( I \) . If \( f \) is defined in a neighbourhood of \( a \) but is not continuous at \( a \), we say that \( f \) has a discontinuity, or is discontinuous, at \( a \) . (1.4.4) Proposition. Let the real-valued functions \( f \) and \( g \) be continuous at a. Then \( f + g, f - g,{fg},\max \{ f, g\} ,\min \{ f, g\} \), and \( \left| f\right| \) are continuous at a. If also \( g\left( x\right) \neq 0 \) for all \( x \) in some neighbourhood of a, then \( f/g \) is continuous at \( a \) . Proof. This is a simple consequence of Proposition (1.4.2). ## (1.4.5) Exercises . 1 Let \( f \) be defined on a neighbourhood of \( a \) . Prove that \( f \) is continuous at \( a \) if and only if it is continuous on both the left and the right at \( a \) . .2 Give the details of the proof of Proposition (1.4.4). Extend this result to deal with continuity on an interval \( I \) . .3 Prove that a polynomial function is continuous on \( \mathbf{R} \) . .4 Let \( p, q \) be polynomial functions, and \( a \) a real number such that \( q\left( a\right) \neq \) 0 . Prove that the rational function \( p/q \) is continuous at \( a \) . .5 Let \( f \) be continuous at \( a \), and let \( g \) be continuous at \( f\left( a\right) \) . Prove that the composite function \( g \circ f \) is continuous at \( a \) . .6 Prove that \( f \) is continuous at the point \( a \in \mathbf{R} \) if and only if \( f \) is sequentially continuous at \( a \), in the sense that \( f\left( {a}_{n}\right) \rightarrow f\left( a\right) \) whenever \( \left( {a}_{n}\right) \) is a sequence of points of the domain of \( f \) that converges to \( a \) . .7 Let \( f \) be defined in an interval \( \left( {a - r, a + r}\right) \) where \( r > 0 \) . The oscillation of \( f \) at \( a \) is \[ \omega \left( {f, a}\right) = \mathop{\lim }\limits_{{\delta \rightarrow 0}}\sup \{ f\left( x\right) - f\left( y\right) : x, y \in \left( {a - \delta, a + \delta }\right) \} . \] Prove that \( f \) is continuous at \( a \) if and only if \( \omega \left( {f, a}\right) = 0 \) . .8 Let \( f \) be an increasing function on \( \left\lbrack {a, b}\right\rbrack \) . Prove that \( f\left( {\xi }^{ - }\right) \) exists for each \( \xi \in (a, b\rbrack \), and that \( f\left( {\xi }^{ + }\right) \) exists for each \( \xi \in \lbrack a, b) \) . By considering the sets \[ \left\{ {x \in \left( {a, b}\right) : \left| {f\left( {x}^{ + }\right) - f\left( {x}^{ - }\right) }\right| > \frac{1}{n}}\right\} , \] with \( n \) a positive integer, prove that the set of points of \( \left\lbrack {a, b}\right\rbrack \) at which \( f \) has a discontinuity is either empty or countable. .9 Let \( {q}_{0},{q}_{1},\ldots \) be a one-one enumeration of \( \mathbf{Q} \cap \left\lbrack {0,1}\right\rbrack \), and for each \( x \in \left\lbrack {0,1}\right\rbrack \) define \[ T\left( x\right) = \left\{ {n \in \mathbf{N} : {q}_{n} \leq x}\right\} . \] Define a mapping \( f : \left\lbrack {0,1}\right\rbrack \rightarrow \mathbf{R} \) by \[ f\left( x\right) = \left\{ \begin{array}{ll} 0 & \text{ if }x = 0 \\ {\left. \mathop{\sum }\limits_{{n \in T\left( x\right) }}\right. }^{{2}^{-n}} & \text{ if }0 < x \leq 1. \end{array}\right. \] Prove that (i) \( f \) is strictly increasing, (ii) \( f \) is continuous at each irrational point of \( \left\lbrack {0,1}\right\rbrack \), and (iii) \( f \) is discontinuous at each rational point of \( \left\lbrack {0,1}\right\rbrack \) . .10 Let \( {\left( {f}_{n}\right) }_{n = 1}^{\infty } \) be a sequence of functions on an interval \( I \), and suppose that there exists a convergent series \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{M}_{n} \) of nonnegative terms such that \( \left| {{f}_{n}\left( x\right) }\right| \leq {M}_{n} \) for each \( x \in I \) and each \( n \) . Prove that for each \( \varepsilon > 0 \) there exists \( N \) such that \( 0 \leq \mathop{\sum }\limits_{{n = j + 1}}^{k}\left| {{f}_{n}\left( x\right) }\right| < \varepsilon \) whenever \( k > j \geq N \) and \( x \in I \) . Hence prove that \( f\left( x\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }{f}_{n}\left( x\right) \) defines a function on \( I \) (Weierstrass’s \( M \) -test). Prove also that if each \( {f}_{n} \) is continuous on \( I \), then so is \( f \) . .11 Give two proofs that exp is a continuous function on \( \mathbf{R} \) . .12 Prove that if \( a > 0 \), then the function \( x \mapsto {a}^{x} \) is continuous on \( \mathbf{R} \) . (Note that \( {a}^{x} = \exp \left( {x\log a}\right) \) .) .13 Prove that if \( a > 0 \), then the function \( x \mapsto {\log }_{a}x \) is continuous on \( \mathbf{R} \) . (First take the case \( a > 1 \) . Given \( x > 0 \) and \( \varepsilon > 0 \), choose a positive integer \( n > 1/\varepsilon \), and then \( \delta \in \left( {0, x}\right) \) such that \( \left( {x + \delta }\right) /x < {a}^{1/n} \) and \( \left( {x - \delta }\right) /x > {a}^{-1/n} \) .) .14 Prove that the functions sin and cos, defined by \[ \sin x = \mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{{\left( -1\right) }^{n}{x}^{{2n} + 1}}{\left( {{2n} + 1}\right) !} \] \[ \cos x = \mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{{\left( -1\right) }^{n}{x}^{2n}}{\left( {2n}\right) !} \] are (well defined and) continuous on \( \mathbf{R} \) . .15 Let \( I \) be the interval of convergence of the power series \( f\left( x\right) = \) \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n} \) . Prove that \( f \) is continuous on \( I \) . .16 Prove that if \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n} \) is a convergent series, then (i) \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n} \) converges for all \( x \in \left( {-1,1}\right) \), and (ii) for each \( \varepsilon > 0 \) there exists \( \delta \in \left( {0,1}\right) \) such that if \( 1 - \sigma < x < 1 \) , then \[ \left| {\mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n} - \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n}}\right| < \varepsilon \] Thus \( \mathop{\lim }\limits_{{x \rightarrow {1}^{ - }}}\mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n} = \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n} \) (Abel’s Limit Theorem). (For (ii), note that \[ \left| {\mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n} - \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n}}\right| \leq \left| {\mathop{\sum }\limits_{{n = 0}}^{N}{a}_{n}\left( {1 - {x}^{n}}\right) }\right| \] \[ + \left| {\mathop{\sum }\limits_{{n = N + 1}}^{\infty }{a}_{n}}\right| + \left| {\mathop{\sum }\limits_{{n = N + 1}}^{\infty }{a}_{n}{x}^{n}}\right| \] for each \( N \) . Use Exercise (1.2.14:15) to handle the last term on the right.) .17 Let \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n},\mathop{\sum }\limits_{{n = 0}}^{\infty }{b}_{n} \) be convergent series with sums \( a, b \), respec-

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< x < 1 \) , then \[ \left| {\mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n} - \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n}}\right| < \varepsilon \] Thus \( \mathop{\lim }\limits_{{x \rightarrow {1}^{ - }}}\mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n} = \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n} \) (Abel’s Limit Theorem). (For (ii), note that \[ \left| {\mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n} - \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n}}\right| \leq \left| {\mathop{\sum }\limits_{{n = 0}}^{N}{a}_{n}\left( {1 - {x}^{n}}\right) }\right| \] \[ + \left| {\mathop{\sum }\limits_{{n = N + 1}}^{\infty }{a}_{n}}\right| + \left| {\mathop{\sum }\limits_{{n = N + 1}}^{\infty }{a}_{n}{x}^{n}}\right| \] for each \( N \) . Use Exercise (1.2.14:15) to handle the last term on the right.) .17 Let \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n},\mathop{\sum }\limits_{{n = 0}}^{\infty }{b}_{n} \) be convergent series with sums \( a, b \), respec- tively, and let \[ {u}_{n} = {a}_{1}{b}_{n} + {a}_{2}{b}_{n - 1} + \cdots + {a}_{n - 1}{b}_{2} + {a}_{n}{b}_{1}. \] Prove that if \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{u}_{n} \) converges, then its sum is \( {ab} \) . (For \( - 1 < x \leq 1 \) set \( f\left( x\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n} \) and \( g\left( x\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{b}_{n}{x}^{n} \) . Then use Exercises (1.2.16:10), (1.2.16:3), and (1.4.5:16).) This is the full form of Cauchy's theorem on the multiplication of series, and should be compared with Exercise (1.2.16:3). Deeper results about continuity indeed, many results in real-variable theory-depend on two fundamental properties of the real line, described in our next two theorems. By a cover of a subset \( S \) of \( \mathbf{R} \) we mean a family \( \mathcal{U} \) of subsets of \( \mathbf{R} \) such that \( S \subset \bigcup \mathcal{U} \) ; we then say that \( S \) is covered by \( \mathcal{U} \) and that \( \mathcal{U} \) covers \( S \) . If also each \( U \in \mathcal{U} \) is an open subset of \( \mathbf{R} \), we refer to \( \mathcal{U} \) as an open cover of \( S \) (in \( \mathbf{R} \) ). By a subcover of a cover \( \mathcal{U} \) of \( S \) we mean a family \( \mathcal{F} \subset \mathcal{U} \) that covers \( S \) ; if also \( \mathcal{F} \) is a finite family, then it is called a finite subcover of \( \mathcal{U} \) . Although there exist shorter proofs of the next theorem (see the next set of exercises), the one we present is adapted to prove a more general result (Theorem (3.3.9)) in Chapter 3. (1.4.6) The Heine-Borel-Lebesgue Theorem. Every open cover of a compact interval \( I \) in \( \mathbf{R} \) contains a finite subcover of \( I \) . Proof. Suppose there exists an open cover \( \mathcal{U} \) of \( I \) that contains no finite subcover of \( I \) . Either the closed right half of \( I \) or the closed left half (or both) cannot be covered by a finite subfamily of \( \mathcal{U} \) : otherwise each half, and therefore \( I \) itself, would be covered by a finite subfamily. Let \( {I}_{1} \) be a closed half of \( I \) that is not covered by a finite subfamily of \( \mathcal{U} \) . In turn, at least one closed half, say \( {I}_{2} \), of \( {I}_{1} \) cannot be covered by a finite subfamily of \( \mathcal{U} \) . Carrying on in this way, we construct a nested sequence \( I \supset {I}_{1} \supset {I}_{2} \supset \cdots \) of closed subintervals of \( I \) such that for each \( n \) , (a) \( \left| {I}_{n}\right| = {2}^{-n}\left| I\right| \) and (b) no finite subfamily of \( \mathcal{U} \) covers \( {I}_{n} \) . By the nested intervals principle (1.2.6), there exists a point \( \xi \in \mathop{\bigcap }\limits_{{n = 1}}^{\infty }{I}_{n} \) . Clearly \( \xi \in I \), so there exists \( U \in \mathcal{U} \) such that \( \xi \in U \) . Since \( U \) is open, there exists \( r > 0 \) such that if \( \left| {x - \xi }\right| < r \), then \( x \in U \) . Using (a), we can find \( N \) such that if \( x \in {I}_{N} \), then \( \left| {x - \xi }\right| < r \) and therefore \( x \in U \) ; thus \( {I}_{N} \subset U \) . This contradicts (b). A real number \( a \) is called a limit point of a subset \( S \) of \( \mathbf{R} \) if each neighbourhood of \( a \) intersects \( S \smallsetminus \{ a\} \) ; or, equivalently, if for each \( \varepsilon > 0 \) there exists \( x \in S \) with \( 0 < \left| {x - a}\right| < \varepsilon \) . By a limit point of a sequence \( \left( {a}_{n}\right) \) we mean a limit point of the set \( \left\{ {{a}_{1},{a}_{2},\ldots }\right\} \) of terms of the sequence. A nonempty subset \( A \) of \( \mathbf{R} \) is said to have the Bolzano-Weierstrass property if each infinite subset \( S \) of \( A \) has a limit point belonging to \( A \) . (1.4.7) The Bolzano-Weierstrass Theorem. Every compact interval in \( \mathbf{R} \) has the Bolzano-Weierstrass property. Proof. Let \( I \) be a compact interval, and \( S \) an infinite subset of \( I \) . By Corollary (1.2.8), any infinite sequence of distinct points of \( S \) contains a convergent subsequence; the limit of that subsequence is a limit point of \( S \) in the closed set \( I \) . ## (1.4.8) Exercises .1 Let \( X \) be a subset of \( \mathbf{R} \) with the Bolzano-Weierstrass property, and let \( \left( {x}_{n}\right) \) be a sequence of points in \( X \) . Show that there exists a subsequence of \( \left( {x}_{n}\right) \) that converges to a limit in \( X \) . (Note Lemma (1.2.7).) .2 Fill in the details of the following alternative proof of the Heine-Borel-Lebesgue Theorem. Let \( \mathcal{U} \) be an open cover of the compact interval \( I = \left\lbrack {a, b}\right\rbrack \), and define \[ A = \{ x \in I : \left\lbrack {a, x}\right\rbrack \text{ is covered by finitely many elements of }\mathcal{U}\} . \] Then \( A \) is nonempty (it contains \( a \) ) and is bounded above; let \( \xi = \) \( \sup A \) . Suppose that \( \xi \neq b \), and derive a contradiction. .3 Fill in the details of the following alternative proof of the Bolzano-Weierstrass Theorem. Suppose the theorem is false; so there exist a compact interval \( I \) and an infinite subset \( S \) of \( I \) such that no limit point of \( S \) belongs to \( I \) . Construct a nested sequence \( I \supset {I}_{1} \supset {I}_{2} \supset \cdots \) of closed subintervals of \( I \) such that for each \( n \) , (a) \( \left| {I}_{n}\right| = {2}^{-n}\left| I\right| \) , (b) \( S \cap {I}_{n} \) is an infinite set, and (c) \( S \cap {I}_{n} \) has no limit points in \( {I}_{n} \) . Let \( \xi \in \mathop{\bigcap }\limits_{{n = 1}}^{\infty }{I}_{n} \), and show that \( \xi \) is a limit point of \( S \) . (This is one of the commonest proofs of the Bolzano-Weierstrass Theorem in textbooks.) .4 Here is a sketch of yet another proof of the Bolzano-Weierstrass Theorem for you to complete. Let \( I \) be a compact interval, and \( S \) an infinite subset of \( I \) ; then the supremum of the set \[ A = \{ x \in I : S \cap \left( {-\infty, x}\right) \text{ is finite or empty }\} \] is a limit point of \( S \) in \( I \) . .5 Let \( S \) be a subset of \( \mathbf{R} \) with the Bolzano-Weierstrass property. Prove that \( S \) is closed and bounded. (For boundedness, use a proof by contradiction.) .6 Show that the Bolzano-Weierstrass Theorem can be proved as a consequence of the Heine-Borel-Lebesgue Theorem. (Let \( I \) be a compact interval in \( \mathbf{R} \), assume the Heine-Borel-Lebesgue Theorem (1.4.6), and suppose that there exists an infinite subset \( S \) of \( I \) that has no limit point in \( I \) . First show that for each \( s \in \bar{S} \) there exists \( {r}_{s} > 0 \) such that \( S \cap \left( {s - {r}_{s}, s + {r}_{s}}\right) = \{ s\} \) .) .7 Let \( f \) be a real-valued function defined on an interval \( I \) . We say that \( f \) is uniformly continuous on \( I \) if to each \( \varepsilon > 0 \) there corresponds \( \delta > 0 \) such that \( \left| {f\left( x\right) - f\left( {x}^{\prime }\right) }\right| < \varepsilon \) whenever \( x,{x}^{\prime } \in I \) and \( \left| {x - {x}^{\prime }}\right| < \delta \) . Show that a uniformly continuous function is continuous. Give an example of \( I \) and \( f \) such that \( f \) is continuous, but not uniformly continuous, on \( I \) . .8 Use the Heine-Borel-Lebesgue Theorem to prove the Uniform Continuity Theorem: a continuous real-valued function \( f \) on a compact interval \( I \subset \mathbf{R} \) is uniformly continuous. (For each \( \varepsilon > 0 \) and each \( x \in I \), choose \( {\delta }_{x} > 0 \) such that if \( {x}^{\prime } \in I \) and \( \left| {x - {x}^{\prime }}\right| < 2{\delta }_{x} \), then \( \left| {f\left( x\right) - f\left( {x}^{\prime }\right) }\right| < \varepsilon /2 \) . The intervals \( \left( {x - {\delta }_{x}, x + {\delta }_{x}}\right) \) form an open cover of \( I \) .) .9 Prove the Uniform Continuity Theorem (see the previous exercise) using the Bolzano-Weierstrass Theorem. (If \( f : I \rightarrow \mathbf{R} \) is not uniformly continuous, then there exists \( \alpha > 0 \) with the following property: for each \( n \in {\mathbf{N}}^{ + } \) there exist \( {x}_{n},{y}_{n} \in I \) such that \( \left| {{x}_{n} - {y}_{n}}\right| < 1/n \) and \( \left. {\left| {f\left( {x}_{n}\right) - f\left( {y}_{n}\right) }\right| \geq \alpha \text{.}}\right) \) The proof of the following result about boundedness of real-valued functions illustrates well the application of the Heine-Borel-Lebesgue Theorem. (1.4.9) Theorem. A continuous real-valued function \( f \) on a compact interval \( I \) is bounded; moreover, \( f \) attains its bounds in the sense that there exist points \( \xi ,\eta \) of \( I \) such that \( f\left( \xi \right) = \inf f \) and \( f\left( \eta \right) = \sup f \) . Proof. For each \( x \in I \) choose \( {\delta }_{x} > 0 \) such that if \( {x}

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ve the Uniform Continuity Theorem (see the previous exercise) using the Bolzano-Weierstrass Theorem. (If \( f : I \rightarrow \mathbf{R} \) is not uniformly continuous, then there exists \( \alpha > 0 \) with the following property: for each \( n \in {\mathbf{N}}^{ + } \) there exist \( {x}_{n},{y}_{n} \in I \) such that \( \left| {{x}_{n} - {y}_{n}}\right| < 1/n \) and \( \left. {\left| {f\left( {x}_{n}\right) - f\left( {y}_{n}\right) }\right| \geq \alpha \text{.}}\right) \) The proof of the following result about boundedness of real-valued functions illustrates well the application of the Heine-Borel-Lebesgue Theorem. (1.4.9) Theorem. A continuous real-valued function \( f \) on a compact interval \( I \) is bounded; moreover, \( f \) attains its bounds in the sense that there exist points \( \xi ,\eta \) of \( I \) such that \( f\left( \xi \right) = \inf f \) and \( f\left( \eta \right) = \sup f \) . Proof. For each \( x \in I \) choose \( {\delta }_{x} > 0 \) such that if \( {x}^{\prime } \in I \) and \( \left| {x - {x}^{\prime }}\right| < {\delta }_{x} \) , then \( \left| {f\left( x\right) - f\left( {x}^{\prime }\right) }\right| < 1 \) . The intervals \( \left( {x - {\delta }_{x}, x + {\delta }_{x}}\right) \), where \( x \in I \), form an open cover of \( I \) . By Theorem (1.4.6), there exist finitely many points \( {x}_{1},\ldots ,{x}_{N} \) of \( I \) such that \[ I \subset \mathop{\bigcup }\limits_{{k = 1}}^{N}\left( {{x}_{k} - {\delta }_{{x}_{k}},{x}_{k} + {\delta }_{{x}_{k}}}\right) . \] Let \[ c = 1 + \max \left\{ {\left| {f\left( {x}_{1}\right) }\right| ,\ldots ,\left| {f\left( {x}_{N}\right) }\right| }\right\} \] and consider any point \( x \in I \) . Choosing \( k \) such that \( x \in \left( {{x}_{k} - {\delta }_{{x}_{k}},{x}_{k} + {\delta }_{{x}_{k}}}\right) \) , we have \[ \left| {f\left( x\right) }\right| \leq \left| {f\left( x\right) - f\left( {x}_{k}\right) }\right| + \left| {f\left( {x}_{k}\right) }\right| \] \[ < 1 + \left| {f\left( {x}_{k}\right) }\right| \] \[ \leq c \] so \( f \) is bounded on \( I \) . Now write \[ m = \inf f,\;M = \sup f. \] Suppose that \( f\left( x\right) \neq M \), and therefore \( f\left( x\right) < M \), for all \( x \in I \) . Then \( x \mapsto 1/\left( {M - f\left( x\right) }\right) \) is a continuous mapping of \( I \) into \( {\mathbf{R}}^{ + } \), by Proposition (1.4.4), and so, by the first part of this proof, has a supremum \( G > 0 \) . For each \( x \in I \) we then have \( M - f\left( x\right) \geq 1/G \) and therefore \( f\left( x\right) \leq M - 1/G \) . This contradicts our choice of \( M \) as the supremum of \( f \) . ## (1.4.10) Exercises .1 Prove both parts of Theorem (1.4.9) using the Bolzano-Weierstrass Theorem and contradiction arguments. .2 Let \( f \) be a continuous function on \( \mathbf{R} \) such that \( f\left( x\right) \rightarrow \infty \) as \( x \rightarrow \pm \infty \) . Prove that there exists \( \xi \in \mathbf{R} \) such that \( f\left( x\right) \geq f\left( \xi \right) \) for all \( x \in \mathbf{R} \) . .3 Let \( f \) be a continuous function on \( \mathbf{R} \) such that \( f\left( x\right) \rightarrow 0 \) as \( x \rightarrow \pm \infty \) . Prove that \( f \) is both bounded and uniformly continuous. (1.4.11) The Intermediate Value Theorem. If \( f \) is a continuous real-valued function on an interval \( I \), then \( f\left( I\right) \) has the intermediate value property (page 36). Proof. Let \( a, b \) be points of \( I \), and \( y \) a real number such that \( f\left( a\right) < y < \) \( f\left( b\right) \) ; without loss of generality assume that \( a < b \) . Then \[ S = \{ x \in \left\lbrack {a, b}\right\rbrack : f\left( x\right) < y\} \] is nonempty (it contains \( a \) ) and bounded above by \( b \), so \( \xi = \sup S \) exists. Note that \( \xi < b \) and that \( (\xi, b\rbrack \subset I \) . We show that \( f\left( \xi \right) = y \) . To this end, suppose first that \( f\left( \xi \right) < y \) . Then, by Exercise (1.4.3:7), there exists \( \delta \in \left( {0, b - \xi }\right) \) such that if \( x \in I \) and \( \left| {x - \xi }\right| < \delta \), then \( f\left( x\right) < y \) ; in particular, \( f\left( x\right) < y \) for all \( x \in \left( {\xi ,\xi + \delta }\right) \), which contradicts the definition of \( \xi \) as the supremum of \( S \) . Thus \( f\left( \xi \right) \geq y \) . Now suppose that \( f\left( \xi \right) > y \) ; then \( \xi > a \) . By another application of Exercise (1.4.3: 7), there exists \( {\delta }^{\prime } \in \left( {0,\xi - a}\right) \) such that if \( \xi - {\delta }^{\prime } < x < \xi \) , then \( f\left( x\right) > y \) . This is impossible, since, by the definition of "supremum", there exist points \( x \) of \( \left( {a,\xi }\right) \) arbitrarily close to \( \xi \) with \( f\left( x\right) < y \) . Hence \( f\left( \xi \right) \leq y \), and therefore \( f\left( \xi \right) = y \) . (1.4.12) Corollary. Let \( f \) be a continuous real-valued function on a compact interval \( I \), and let \( m = \inf f, M = \sup f \) . Then \( f\left( I\right) = \left\lbrack {m, M}\right\rbrack \) . Proof. Use Theorems (1.4.9) and (1.4.11). ## (1.4.13) Exercises .1 Fill in the details of the following common proof of the Intermediate Value Theorem. Let \( f\left( a\right) < y < f\left( b\right) \), write \( {a}_{0} = a,{b}_{0} = \) \( b,{c}_{0} = \frac{1}{2}\left( {{a}_{0} + {b}_{0}}\right) \), and assume without loss of generality that \( a < b \) . If \( f\left( {c}_{0}\right) = y \), there is nothing to prove and we stop our construction. Otherwise, by repeated interval-halving, we construct points \( {a}_{0},{b}_{0},{c}_{0},{a}_{1},{b}_{1},{c}_{1},\ldots \) such that - either \( f\left( {c}_{n}\right) = 0 \) for some \( n \) and the construction stops, - or else the construction proceeds ad infinitum, \( {a}_{n} \leq {a}_{n + 1} \leq \) \( {b}_{n + 1} \leq {b}_{n}, f\left( {a}_{n}\right) < y, f\left( {b}_{n}\right) > y,{c}_{n} = \frac{1}{2}\left( {{a}_{n} + {b}_{n}}\right) \), and \[ 0 < {b}_{n} - {a}_{n} = {\left( \frac{1}{2}\right) }^{n}\left( {b - a}\right) . \] Choosing \( x \in \mathop{\bigcap }\limits_{{n = 1}}^{\infty }\left\lbrack {{a}_{n},{b}_{n}}\right\rbrack \), we now show that \( f\left( x\right) = 0 \) . 2 Use the Intermediate Value Theorem to prove that if \( b > 0 \) and \( n \) is an odd positive integer, then \( b \) has an \( n \) th root - that is, there exists \( r \in \mathbf{R} \) such that \( {r}^{n} = b \) . (Of course, this result also follows from our definition of \( {a}^{x} \) in Section 1; but it is instructive to see how it can be derived by other means, such as the Intermediate Value Theorem.) .3 Show that any polynomial equation \[ {x}^{n} + {a}_{n - 1}{x}^{n - 1} + \cdots + {a}_{1}x + {a}_{0} = 0 \] of odd degree \( n \), with coefficients \( {a}_{k} \in \mathbf{R} \), has at least one real solution. .4 What can you say about a function \( f \) that is continuous on \( \left\lbrack {0,1}\right\rbrack \) and assumes only rational values? .5 Let \( f, g \) be continuous functions on \( \left\lbrack {0,1}\right\rbrack \) such that \( f\left( x\right) \in \left\lbrack {0,1}\right\rbrack \) for all \( x, g\left( 0\right) = 0 \), and \( g\left( 1\right) = 1 \) . Show that \( f\left( x\right) = g\left( x\right) \) for some \( x \in \left\lbrack {0,1}\right\rbrack \) . .6 Prove that there is no continuous function \( f : \mathbf{R} \rightarrow \mathbf{R} \) that assumes each real value exactly twice. .7 Let \( f \) be continuous and one-one on an interval \( I \) ; then \( f\left( I\right) \) is an interval, by Corollary (1.4.12). Prove that (i) either \( f \) is strictly increasing on \( I \) or else \( f \) is strictly decreasing on \( I \) ; (ii) if \( a \in {I}^{ \circ } \), then \( f\left( a\right) \in f{\left( I\right) }^{ \circ } \) ; (iii) \( {f}^{-1} \) is continuous on \( f\left( I\right) \) . (For (iii), show that \( f \) is sequentially continuous at each point of \( f\left( I\right) \) . You will need Corollary (1.2.8), Exercise (1.4.5: 6), and Exercise (1.2.9: 4).) Although the Intermediate Value Theorem has many applications, especially in the solution of equations, none of its proofs provides an algorithm for constructing the point \( x \) with \( f\left( x\right) = y \) . This claim may come as a surprise: for is not the interval-halving proof in Exercise (1.4.13:1) algorithmic? Alas, it is not: for, as any good computer scientist knows, there is no algorithm that enables us to decide, for given real numbers \( y \) and \( z \) , whether \( y = z \) or \( y \neq z \) . (For further discussion of these matters, see the Prolog of [5], and pages 65-66 of [8].) ## 1.5 Calculus In this section we cover the fundamentals of the differential and integral calculus of functions of one real variable. We do so rapidly, leaving many details to the exercises, on the assumption that you will have seen much of the material in elementary calculus courses. Let \( I \) be an interval in \( \mathbf{R},{x}_{0} \) a point of \( I \), and \( f \) a real-valued function whose domain includes \( I \) . We say that \( f \) is - differentiable on the left at \( {x}_{0} \) if its left-hand derivative at \( {x}_{0} \) , \[ {f}^{\prime }\left( {x}_{0}^{ - }\right) = \mathop{\lim }\limits_{{x \rightarrow {x}_{0}^{ - }}}\frac{f\left( x\right) - f\left( {x}_{0}\right) }{x - {x}_{0}} = \mathop{\lim }\limits_{{h \rightarrow {0}^{ - }}}\frac{f\left( {{x}_{0} + h}\right) - f\left( {x}_{0}\right) }{h}, \] exists; - differentiable on the right at \( {x}_{0} \) if its right-hand derivative at \( {x}_{0} \) , \[ {f}^{\prime }\left( {x}_{0}^{ + }\right) = \mathop{\lim }\limits_{{x \rightarrow {x}_{0}^{ + }}}\frac{f\left( x\right) - f\left( {x}_{0}\right) }{x - {x}_{0}} = \mathop{\lim }\limits_{{h \rightarrow {0}^{ + }}}\frac{f\left( {{x}_{0} + h}\right) - f\left( {x}_{0}\right) }{h}, \] exists; - differentiable a

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material in elementary calculus courses. Let \( I \) be an interval in \( \mathbf{R},{x}_{0} \) a point of \( I \), and \( f \) a real-valued function whose domain includes \( I \) . We say that \( f \) is - differentiable on the left at \( {x}_{0} \) if its left-hand derivative at \( {x}_{0} \) , \[ {f}^{\prime }\left( {x}_{0}^{ - }\right) = \mathop{\lim }\limits_{{x \rightarrow {x}_{0}^{ - }}}\frac{f\left( x\right) - f\left( {x}_{0}\right) }{x - {x}_{0}} = \mathop{\lim }\limits_{{h \rightarrow {0}^{ - }}}\frac{f\left( {{x}_{0} + h}\right) - f\left( {x}_{0}\right) }{h}, \] exists; - differentiable on the right at \( {x}_{0} \) if its right-hand derivative at \( {x}_{0} \) , \[ {f}^{\prime }\left( {x}_{0}^{ + }\right) = \mathop{\lim }\limits_{{x \rightarrow {x}_{0}^{ + }}}\frac{f\left( x\right) - f\left( {x}_{0}\right) }{x - {x}_{0}} = \mathop{\lim }\limits_{{h \rightarrow {0}^{ + }}}\frac{f\left( {{x}_{0} + h}\right) - f\left( {x}_{0}\right) }{h}, \] exists; - differentiable at \( {x}_{0} \) if \( {x}_{0} \) is an interior point of \( I \) and the derivative of \( f \) at \( {x}_{0} \) , \[ {f}^{\prime }\left( {x}_{0}\right) = \mathop{\lim }\limits_{{x \rightarrow {x}_{0}}}\frac{f\left( x\right) - f\left( {x}_{0}\right) }{x - {x}_{0}} = \mathop{\lim }\limits_{{h \rightarrow 0}}\frac{f\left( {{x}_{0} + h}\right) - f\left( {x}_{0}\right) }{h}, \] exists. It follows from Exercise (1.4.3: 2) that \( f \) is differentiable at an interior point \( {x}_{0} \) of its domain if and only if \( {f}^{\prime }\left( {x}_{0}^{ - }\right) \) and \( {f}^{\prime }\left( {x}_{0}^{ + }\right) \) exist and are equal, in which case their common value is \( {f}^{\prime }\left( {x}_{0}\right) \) . We say that \( f \) is differentiable on the interval \( I \) if it is - differentiable at each interior point of \( I \) , - differentiable on the right at the left endpoint of \( I \) if that point belongs to \( I \), and - differentiable on the left at the right endpoint of \( I \) if that point belongs to \( I \) . Higher-order derivatives of \( f \) are defined inductively, as follows. \[ {f}^{\left( 0\right) } = f \] \[ {f}^{\left( 1\right) } = {f}^{\prime } \] \[ {f}^{\left( 2\right) } = {f}^{\prime \prime } = {\left( {f}^{\prime }\right) }^{\prime } \] \[ {f}^{\left( 3\right) } = {f}^{\prime \prime \prime } = {\left( {f}^{\prime \prime }\right) }^{\prime } \] \[ {f}^{\left( n + 1\right) } = {\left( {f}^{\left( n\right) }\right) }^{\prime }\;\left( {n \geq 3}\right) . \] If the \( n \) th derivative \( {f}^{\left( n\right) }\left( x\right) \) exists, then \( f \) is said to be \( n \) -times differentiable at \( x \) ; if \( {f}^{\left( n\right) }\left( x\right) \) exists for each positive integer \( n \), then \( f \) is said to be infinitely differentiable at \( x \) . Definitions of notions such as \( n \) th right-hand derivative, n-times differentiable on an interval, and infinitely differentiable on an interval are formulated analogously. ## (1.5.1) Exercises .1 Prove that if \( f \) is differentiable at \( {x}_{0} \), then it is continuous at \( {x}_{0} \) . Give an example of a function \( f : \mathbf{R} \rightarrow \mathbf{R} \) such that \( {f}^{\prime }\left( {0}^{ - }\right) \) and \( {f}^{\prime }\left( {0}^{ + }\right) \) both exist but \( f \) is not continuous at 0 . .2 For each \( x \in \mathbf{R} \) write \[ \rho \left( {x,\mathbf{Z}}\right) = \inf \{ \left| {x - n}\right| : n \in \mathbf{Z}\} \] and \[ f\left( x\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{\rho \left( {{10}^{n}x,\mathbf{Z}}\right) }{{10}^{n}}. \] Prove that \( f \) is continuous, but nowhere differentiable, on \( \mathbf{R} \) . (For continuity use Exercise (1.4.5:10). To show that \( f \) is not differentiable at \( x \), it is enough to take \( 0 \leq x < 1 \) . Let \( 0.{d}_{1}{d}_{2}\ldots \) be a decimal expansion of \( x \), the terminating expansion if there is one. Define \( {h}_{k} \) to be \( - {10}^{-k} \) if \( {a}_{k} = 4 \) or 9, and \( {10}^{-k} \) otherwise, and consider \( {h}_{k}^{-1}\left( {f\left( {x + {h}_{k}}\right) - f\left( x\right) }\right) \) .) This example is due to van der Waerden [54]. Weierstrass, in a lecture to the Berlin Academy in 1872, gave the first example of a continuous, nowhere differentiable function: namely, \[ f\left( x\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}^{n}\cos \left( {{b}^{n}{\pi x}}\right) \] where \( 0 < a < 1, b \) is an odd positive integer, and \( {ab} > 1 + {3\pi }/2 \) ; for a discussion of a special case of Weierstrass's example, see [28], pages 38-41. .3 Prove that if \( f \) is differentiable at \( x \), then \[ {f}^{\prime }\left( x\right) = \mathop{\lim }\limits_{{h, k \rightarrow {0}^{ + }}}\frac{f\left( {x + h}\right) - f\left( {x - k}\right) }{h + k}. \] Give an example of a function \( f \) where \( \mathop{\lim }\limits_{{h \rightarrow 0}}\left( {\left( {f\left( h\right) - f\left( {-h}\right) }\right) /{2h}}\right) \) exists but \( f \) is not differentiable at 0 . .4 Let \( f\left( x\right) = {x}^{n} \), where \( n \) is an integer. Using the definition of "differentiable", prove that \( {f}^{\prime }\left( x\right) = n{x}^{n - 1} \) for all \( x \in \mathbf{R} \) . .5 Let \( f \) and \( g \) be differentiable at \( a \) . Prove that \( f + g, f - g,{cf}\left( {c \in \mathbf{R}}\right) \) , and \( {fg} \) are differentiable at \( a \), and that \[ {\left( f + g\right) }^{\prime }\left( a\right) = {f}^{\prime }\left( a\right) + {g}^{\prime }\left( a\right) \] \[ {\left( f - g\right) }^{\prime }\left( a\right) = {f}^{\prime }\left( a\right) - {g}^{\prime }\left( a\right) , \] \[ {\left( cf\right) }^{\prime }\left( a\right) = c{f}^{\prime }\left( a\right) \] \[ {\left( fg\right) }^{\prime }\left( a\right) = f\left( a\right) {g}^{\prime }\left( a\right) + {f}^{\prime }\left( a\right) g\left( a\right) . \] .6 Under the conditions of the last exercise, suppose also that \( g\left( a\right) \neq 0 \) . Give two proofs that \( f/g \) is differentiable at \( a \), and that \[ {\left( \frac{f}{g}\right) }^{\prime }\left( a\right) = \frac{{f}^{\prime }\left( a\right) g\left( a\right) - f\left( a\right) {g}^{\prime }\left( a\right) }{g{\left( a\right) }^{2}}. \] .7 Using the exponential series, prove that \( {\exp }^{\prime }\left( 0\right) = 1 \) . Hence prove that \( {\exp }^{\prime }\left( x\right) = \exp \left( x\right) \) for all \( x \in \mathbf{R} \) . Our next proposition, the Chain Rule, is possibly the most troublesome result of elementary calculus. (1.5.2) Proposition. If \( f \) is differentiable at \( a \), and \( g \) is differentiable at \( f\left( a\right) \), then \( g \circ f \) is differentiable at \( a \), and \[ {\left( g \circ f\right) }^{\prime }\left( a\right) = {g}^{\prime }\left( {f\left( a\right) }\right) \cdot {f}^{\prime }\left( a\right) . \] Proof. Setting \( b = f\left( a\right) \), define \[ h\left( u\right) = \left\{ \begin{array}{ll} \frac{g\left( u\right) - g\left( b\right) }{u - b} & \text{ if }u \neq b \\ {g}^{\prime }\left( b\right) & \text{ if }u = b. \end{array}\right. \] For all \( x \neq a \) in some neighbourhood of \( a \) we have \[ \frac{g\left( {f\left( x\right) }\right) - g\left( {f\left( a\right) }\right) }{x - a} = \left( {h \circ f}\right) \left( x\right) \cdot \frac{f\left( x\right) - f\left( a\right) }{x - a}. \] (1) (Note that in verifying this identity, we must consider the possibility that \( f\left( x\right) = f\left( a\right) \) .) Since \( g \) is differentiable at \( b, h \) is continuous at \( b \) . Moreover, \( f \) is differentiable, and therefore (by Exercise (1.5.1:1)) continuous, at \( a \) ; so \( h \circ f \) is continuous at \( a \), by Exercise (1.4.5: 5). Hence \[ \mathop{\lim }\limits_{{x \rightarrow a}}\left( {\left( {h \circ f}\right) \left( x\right) \cdot \frac{f\left( x\right) - f\left( a\right) }{x - a}}\right) = \mathop{\lim }\limits_{{x \rightarrow a}}\left( {h \circ f}\right) \left( x\right) \cdot \mathop{\lim }\limits_{{x \rightarrow a}}\frac{f\left( x\right) - f\left( a\right) }{x - a} \] \[ = \left( {h \circ f}\right) \left( a\right) \cdot {f}^{\prime }\left( a\right) \] \[ = {g}^{\prime }\left( b\right) {f}^{\prime }\left( a\right) \text{.} \] The result now follows immediately from (1). (1.5.3) Proposition. Let \( I \) be an open interval, \( f \) a one-one continuous function on \( I \), and \( a \in I \), such that \( {f}^{\prime }\left( a\right) \) exists and is nonzero. Then the inverse function \( {f}^{-1} \) is differentiable at \( f\left( a\right) \), and \[ {\left( {f}^{-1}\right) }^{\prime }\left( {f\left( a\right) }\right) = \frac{1}{{f}^{\prime }\left( a\right) }. \] Proof. Note that \( J = f\left( I\right) \) is an interval, by Theorem (1.4.11) and Proposition (1.3.3); moreover, by Exercise (1.4.13:7), \( f\left( a\right) \) is an interior point of \( J \) and \( {f}^{-1} \) is continuous on \( J \) . Let \( \left( {y}_{n}\right) \) be any sequence in \( J \smallsetminus \{ f\left( a\right) \} \) that converges to \( f\left( a\right) \), and write \( {x}_{n} = {f}^{-1}\left( {y}_{n}\right) \) ; then \( {x}_{n} \neq a \) and \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}{f}^{-1}\left( {x}_{n}\right) = {f}^{-1}\left( {f\left( a\right) }\right) = a. \] Since \( f \) is one-one, it follows that \[ \frac{f\left( {x}_{n}\right) - f\left( a\right) }{{x}_{n} - a} \neq 0 \] whence \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{{f}^{-1}\left( {y}_{n}\right) - {f}^{-1}\left( {f\left( a\right) }\right) }{{y}_{n} - f\left( a\right) } = \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{{x}_{n} - a}{f\left( {x}_{n}\right) - f\left( a\right) } = \frac{1}{{f}^{\prime }\left( a\right) }. \] The desired conclusion now follows from Exercise (1.4.3: 6). ## (1.5.4) Exercises .1 Prove that \( {\log }^{\prime }\left( x\right) = 1/x \) for each \( x > 0 \) . .2 Let \( f\left( x\right) = {x}^{r} \), where \(

1008_(GTM174)Foundations of Real and Abstract Analysis

ht) \) be any sequence in \( J \smallsetminus \{ f\left( a\right) \} \) that converges to \( f\left( a\right) \), and write \( {x}_{n} = {f}^{-1}\left( {y}_{n}\right) \) ; then \( {x}_{n} \neq a \) and \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}{f}^{-1}\left( {x}_{n}\right) = {f}^{-1}\left( {f\left( a\right) }\right) = a. \] Since \( f \) is one-one, it follows that \[ \frac{f\left( {x}_{n}\right) - f\left( a\right) }{{x}_{n} - a} \neq 0 \] whence \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{{f}^{-1}\left( {y}_{n}\right) - {f}^{-1}\left( {f\left( a\right) }\right) }{{y}_{n} - f\left( a\right) } = \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{{x}_{n} - a}{f\left( {x}_{n}\right) - f\left( a\right) } = \frac{1}{{f}^{\prime }\left( a\right) }. \] The desired conclusion now follows from Exercise (1.4.3: 6). ## (1.5.4) Exercises .1 Prove that \( {\log }^{\prime }\left( x\right) = 1/x \) for each \( x > 0 \) . .2 Let \( f\left( x\right) = {x}^{r} \), where \( r \in \mathbf{R} \) . Prove that \( {f}^{\prime }\left( x\right) = r{x}^{r - 1} \) . (Note that \( \left. {{x}^{r} = \exp \left( {r\log x}\right) \text{.}}\right) \) .3 Let \( f \) be a strictly increasing function on an interval \( I \), and let \( a \) be a point of \( I \) such that \( {f}^{\prime }\left( {a}^{ + }\right) \) exists and is nonzero. Prove that the inverse function \( {f}^{-1} \) is differentiable on the right at \( f\left( a\right) \), and that \[ {\left( {f}^{-1}\right) }^{\prime }\left( {f{\left( a\right) }^{ + }}\right) = \frac{1}{{f}^{\prime }\left( {a}^{ + }\right) }. \] .4 Let \( f \) be continuous on the compact interval \( I = \left\lbrack {a, b}\right\rbrack \) and differentiable on \( \left( {a, b}\right) \) . Prove that if \( \xi \in \left( {a, b}\right) \) and \( f\left( \xi \right) = \inf f \), then \( {f}^{\prime }\left( \xi \right) = 0 \) . Hence prove that if \( f\left( a\right) = f\left( b\right) \), then there exists \( \xi \in \left( {a, b}\right) \) such that \( {f}^{\prime }\left( \xi \right) = 0 \) (Rolle’s Theorem). .5 Let \( f \) be continuous on the compact interval \( \left\lbrack {a, b}\right\rbrack \) and \( n \) -times differentiable on \( \left( {a, b}\right) \) . Suppose that there exist \( n + 1 \) distinct points \( x \) of \( \left( {a, b}\right) \) at which \( f\left( x\right) = 0 \) . Show that \( {f}^{\left( n\right) }\left( x\right) = 0 \) for some \( x \in \left( {a, b}\right) \) . .6 Use Rolle’s Theorem to prove the Mean Value Theorem: if \( f \) is continuous on the compact interval \( \left\lbrack {a, b}\right\rbrack \) and differentiable on \( \left( {a, b}\right) \), then there exists \( \xi \in \left( {a, b}\right) \) such that \( f\left( b\right) - f\left( a\right) = {f}^{\prime }\left( \xi \right) \left( {b - a}\right) \) . .7 Let \( f \) be differentiable on an interval \( I \) . Prove that (i) if \( {f}^{\prime }\left( x\right) \geq 0 \) for all \( x \in I \), then \( f \) is increasing on \( I \) ; (ii) if \( {f}^{\prime }\left( x\right) > 0 \) for all \( x \in I \), then \( f \) is strictly increasing on \( I \) ; (iii) if \( {f}^{\prime }\left( x\right) = 0 \) for all \( x \in I \), then \( f \) is constant on \( I \) . .8 Let \( f \) be differentiable on an interval \( I \), with \( {f}^{\prime }\left( x\right) \neq 0 \) for all \( x \in I \) . Prove that \( f \) is one-one, and that either \( {f}^{\prime }\left( x\right) \geq 0 \) for all \( x \in I \) or else \( {f}^{\prime }\left( x\right) \leq 0 \) for all \( x \in I \) . .9 Prove that if \( f \) is differentiable on an interval \( I \), then the range of \( {f}^{\prime } \) has the intermediate value property on \( I \) . (Let \( {f}^{\prime }\left( {x}_{1}\right) < y < {f}^{\prime }\left( {x}_{2}\right) \) , consider \( g\left( x\right) = f\left( x\right) - {yx} \), and use the preceding exercise.) .10 Prove Cauchy’s Mean Value Theorem: if \( f, g \) are continuous on \( \left\lbrack {a, b}\right\rbrack \) and differentiable on \( \left( {a, b}\right) \), then there exists \( \xi \in \left( {a, b}\right) \) such that \[ \left( {f\left( b\right) - f\left( a\right) }\right) {g}^{\prime }\left( \xi \right) = \left( {g\left( b\right) - g\left( a\right) }\right) {f}^{\prime }\left( \xi \right) . \] (Consider the function \( x \mapsto \left( {f\left( b\right) - f\left( a\right) }\right) g\left( x\right) - \left( {g\left( b\right) - g\left( a\right) }\right) f\left( x\right) \) .) .11 Let \( f, g \) be continuous on \( \left\lbrack {a, b}\right\rbrack \) and differentiable on \( \left( {a, b}\right) \), let \( {x}_{0} \in \) \( \left\lbrack {a, b}\right\rbrack \), and suppose that (i) \( {g}^{\prime }\left( x\right) \neq 0 \) for all \( x \neq {x}_{0} \) , (ii) \( f\left( {x}_{0}\right) = g\left( {x}_{0}\right) = 0 \) , (iii) \( l = \mathop{\lim }\limits_{{x \rightarrow {x}_{0}}}\left( {{f}^{\prime }\left( x\right) /{g}^{\prime }\left( x\right) }\right) \) exists. Prove that \( \mathop{\lim }\limits_{{x \rightarrow {x}_{0}}}\left( {f\left( x\right) /g\left( x\right) }\right) = l \) . (l’Hôpital’s Rule. Use the preceding exercise to show that if \( \left( {x}_{n}\right) \) is any sequence in \( \left\lbrack {a, b}\right\rbrack \smallsetminus \left\{ {x}_{0}\right\} \) that converges to \( {x}_{0} \), then \( f\left( {x}_{n}\right) /g\left( {x}_{n}\right) \rightarrow l \) as \( n \rightarrow \infty \) .) .12 Let \( g \) be twice differentiable at 0, with \( g\left( 0\right) = {g}^{\prime }\left( 0\right) = 0 \) . Find \( {f}^{\prime }\left( 0\right) \) , where \( f \) is defined by \[ f\left( x\right) = \left\{ \begin{array}{ll} \frac{g\left( x\right) }{x} & \text{ if }x \neq 0 \\ 0 & \text{ if }x = 0 \end{array}\right. \] The following generalisation of the Mean Value Theorem is one of the most useful results of the differential calculus. Unfortunately, it seems to have no completely transparent, natural proof; all the proofs in the literature use some trick or other to obtain the desired conclusion. (1.5.5) Taylor’s Theorem. Let \( f \) be \( \left( {N + 1}\right) \) -times differentiable on an interval \( I \), and let \( a \in I \) . Then for each \( x \in I \) there exists \( \xi \) between a and \( x \) such that \[ f\left( x\right) = \mathop{\sum }\limits_{{n = 0}}^{N}\frac{{f}^{\left( n\right) }\left( a\right) }{n!}{\left( x - a\right) }^{n} + \frac{{f}^{\left( N + 1\right) }\left( \xi \right) }{N!}{\left( x - \xi \right) }^{N}\left( {x - a}\right) . \] Proof. Fixing \( x \in I \), without loss of generality we take \( a < x \) . Consider the function \( g : \left\lbrack {a, x}\right\rbrack \rightarrow \mathbf{R} \) defined by \[ g\left( t\right) = f\left( x\right) - \mathop{\sum }\limits_{{n = 0}}^{N}\frac{{f}^{\left( n\right) }\left( t\right) }{n!}{\left( x - t\right) }^{n}. \] Using Exercises (1.5.1: 4 and 5), we have \[ {g}^{\prime }\left( t\right) = - {f}^{\prime }\left( t\right) - \mathop{\sum }\limits_{{n = 1}}^{N}\left( {\frac{{f}^{\left( n + 1\right) }\left( t\right) }{n!}{\left( x - t\right) }^{n} - \frac{{f}^{\left( n\right) }\left( t\right) }{\left( {n - 1}\right) !}{\left( x - t\right) }^{n - 1}}\right) \] \[ = - \frac{{f}^{\left( N + 1\right) }\left( t\right) }{N!}{\left( x - t\right) }^{N}. \] Applying the Mean Value Theorem (Exercise (1.5.4: 6)), we obtain \( \xi \in \) \( \left( {a, x}\right) \) such that \[ - \frac{{f}^{\left( N + 1\right) }\left( \xi \right) }{N!}{\left( x - \xi \right) }^{N} = {g}^{\prime }\left( \xi \right) = \frac{g\left( x\right) - g\left( a\right) }{x - a}. \] Then, as \( g\left( x\right) = 0 \) , \[ g\left( a\right) = \frac{{f}^{\left( N + 1\right) }\left( \xi \right) }{N!}{\left( x - \xi \right) }^{N}\left( {x - a}\right) , \] which is equivalent to the desired result. The expression \[ \mathop{\sum }\limits_{{n = 0}}^{N}\frac{{f}^{\left( n\right) }\left( a\right) }{n!}{\left( x - a\right) }^{n} \] is called the Taylor polynomial of degree \( n \) at \( a \), and \[ f\left( x\right) - \mathop{\sum }\limits_{{n = 0}}^{N}\frac{{f}^{\left( n\right) }\left( a\right) }{n!}{\left( x - a\right) }^{n} \] is called the remainder term of order \( n \) in Taylor’s Theorem. The theorem, as stated, has the remainder term in the Cauchy form: \[ \frac{{f}^{\left( N + 1\right) }\left( \xi \right) }{N!}{\left( x - \xi \right) }^{N}\left( {x - a}\right) \] The next corollary gives us an alternative form-the Lagrange form-of the remainder. (1.5.6) Corollary. Under the hypotheses of Taylor’s Theorem, for each \( x \in I \) there exists \( t \) between \( a \) and \( x \) such that \[ f\left( x\right) = \mathop{\sum }\limits_{{n = 0}}^{N}\frac{{f}^{\left( n\right) }\left( a\right) }{n!}{\left( x - a\right) }^{n} + \frac{{f}^{\left( N + 1\right) }\left( t\right) }{\left( {N + 1}\right) !}{\left( x - a\right) }^{N + 1}. \] Proof. Again we take \( a < x \) and use a trick. With \( g \) the function introduced in the preceding proof of Taylor's Theorem, we apply Cauchy's Mean Value Theorem (Exercise (1.5.4:10)) to \( g \) and \( t \mapsto {\left( x - t\right) }^{N + 1} \) . Since \( g\left( x\right) = 0 \), this yields \( t \in \left( {a, x}\right) \) such that \[ \frac{g\left( a\right) }{{\left( x - a\right) }^{N + 1}} = \frac{\frac{{f}^{\left( N + 1\right) }\left( t\right) }{N!}{\left( x - t\right) }^{N}}{\left( {N + 1}\right) {\left( x - t\right) }^{N}}. \] Hence \[ g\left( a\right) = \frac{{f}^{\left( N + 1\right) }\left( t\right) }{\left( {N + 1}\right) !}{\left( x - a\right) }^{N + 1}, \] from which the required conclusion follows. (1.5.7) Proposition. Let \( I \) be the interval of convergence of the power series \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n} \), and let \( f\left( x\right) \) be the sum of the series on \( I \) . Then \( f \) is differentiable, and \( {f}^{\prime }\left( x\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }n{a}_{n}{x}^{n - 1} \), at each interior point of \( I \) . Proof. We first r

1008_(GTM174)Foundations of Real and Abstract Analysis

ly Cauchy's Mean Value Theorem (Exercise (1.5.4:10)) to \( g \) and \( t \mapsto {\left( x - t\right) }^{N + 1} \) . Since \( g\left( x\right) = 0 \), this yields \( t \in \left( {a, x}\right) \) such that \[ \frac{g\left( a\right) }{{\left( x - a\right) }^{N + 1}} = \frac{\frac{{f}^{\left( N + 1\right) }\left( t\right) }{N!}{\left( x - t\right) }^{N}}{\left( {N + 1}\right) {\left( x - t\right) }^{N}}. \] Hence \[ g\left( a\right) = \frac{{f}^{\left( N + 1\right) }\left( t\right) }{\left( {N + 1}\right) !}{\left( x - a\right) }^{N + 1}, \] from which the required conclusion follows. (1.5.7) Proposition. Let \( I \) be the interval of convergence of the power series \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n} \), and let \( f\left( x\right) \) be the sum of the series on \( I \) . Then \( f \) is differentiable, and \( {f}^{\prime }\left( x\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }n{a}_{n}{x}^{n - 1} \), at each interior point of \( I \) . Proof. We first recall from Exercise (1.2.16:16) that the power series \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n} \) and \( \mathop{\sum }\limits_{{n = 1}}^{\infty }n{a}_{n}{x}^{n - 1} \) have the same radius of convergence \( R \) . Given \( x \in {I}^{ \circ } \), let \[ r = \frac{1}{2}\left( {\left| x\right| + R}\right) \] Then \( \mathop{\sum }\limits_{{n = 1}}^{\infty }n{a}_{n}{r}^{n - 1} \) converges absolutely, by Exercise (1.2.16:10). Given \( \varepsilon > 0 \), choose \( N \) such that \( \mathop{\sum }\limits_{{n = N + 1}}^{\infty }n\left| {a}_{n}\right| {r}^{n - 1} < \varepsilon \) . If \( h \neq 0 \) and \( \left| {x + h}\right| < \) \( r \), then, using the Mean Value Theorem, for each \( n \geq N + 1 \) we obtain \( {\theta }_{n} \) between 0 and \( h \) such that \[ \frac{{\left( x + h\right) }^{n} - {x}^{n}}{h} = n{\left( x + {\theta }_{n}\right) }^{n - 1}. \] 60 Hence \[ \left| {\mathop{\sum }\limits_{{n = N + 1}}^{\infty }{a}_{n}\left( {\frac{{\left( x + h\right) }^{n} - {x}^{n}}{h} - n{x}^{n - 1}}\right) }\right| \] \[ \leq \mathop{\sum }\limits_{{n = N + 1}}^{\infty }\left| {a}_{n}\right| \left| {\frac{{\left( x + h\right) }^{n} - {x}^{n}}{h} - n{x}^{n - 1}}\right| \] \[ \leq \mathop{\sum }\limits_{{n = N + 1}}^{\infty }n\left| {a}_{n}\right| \left| {{\left( x + {\theta }_{n}\right) }^{n - 1} - {x}^{n - 1}}\right| \] \[ \leq \mathop{\sum }\limits_{{n = N + 1}}^{\infty }n\left| {a}_{n}\right| \left( {{\left| x + {\theta }_{n}\right| }^{n - 1} + {\left| x\right| }^{n - 1}}\right) \] \[ \leq 2\mathop{\sum }\limits_{{n = N + 1}}^{\infty }n\left| {a}_{n}\right| {r}^{n - 1} \] \[ < {2\varepsilon }\text{.} \] It follows that \[ \left| {\frac{f\left( {x + h}\right) - f\left( x\right) }{h} - \mathop{\sum }\limits_{{n = 1}}^{\infty }n{a}_{n}{x}^{n - 1}}\right| \] \[ \leq \left| {\mathop{\sum }\limits_{{n = 1}}^{N}{a}_{n}\left( {\frac{{\left( x + h\right) }^{n} - {x}^{n}}{h} - n{x}^{n - 1}}\right) }\right| \] \[ + \left| {\mathop{\sum }\limits_{{n = N + 1}}^{\infty }{a}_{n}\left( {\frac{{\left( x + h\right) }^{n} - {x}^{n}}{h} - n{x}^{n - 1}}\right) }\right| \] \[ < \mathop{\sum }\limits_{{n = 1}}^{N}\left| {a}_{n}\right| \left| {\frac{{\left( x + h\right) }^{n} - {x}^{n}}{h} - n{x}^{n - 1}}\right| + {2\varepsilon }. \] Now, there exists \( \delta > 0 \) such that if \( 0 < \left| h\right| < \delta \), then \( \left| {x + h}\right| < r \) and \[ \left| {\frac{{\left( x + h\right) }^{n} - {x}^{n}}{h} - n{x}^{n - 1}}\right| < {\left( 1 + \mathop{\sum }\limits_{{n = 1}}^{N}\left| {a}_{n}\right| \right) }^{-1}\varepsilon \;\left( {1 \leq n \leq N}\right) . \] For such \( h \) we then have \[ \left| {\frac{f\left( {x + h}\right) - f\left( x\right) }{h} - \mathop{\sum }\limits_{{n = 1}}^{\infty }n{a}_{n}{x}^{n - 1}}\right| < {3\varepsilon }. \] Since \( \varepsilon > 0 \) is arbitrary, the result follows. ## (1.5.8) Exercises .1 Let \( f \) be infinitely differentiable on an interval \( I \), and suppose that there exists \( M > 0 \) such that \( \left| {{f}^{\left( n\right) }\left( x\right) }\right| \leq M \) for all sufficiently large \( n \) and all \( x \in I \) . Given \( a \in I \), prove that the Taylor expansion, or Taylor series, of \( f \) about \( a \) , \[ \mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{{f}^{\left( n\right) }\left( a\right) }{n!}{\left( x - a\right) }^{n} \] converges to \( f\left( x\right) \) for each \( x \in I \) . .2 Find the Taylor expansion of \( \exp \left( {-{x}^{2}}\right) \) about 0 . For what values of \( x \) does this expansion converge to \( \exp \left( {-{x}^{2}}\right) \) ? .3 Prove that \( f\left( x\right) = \exp \left( x\right) \) defines the unique differentiable function such that \( f\left( 0\right) = 1 \) and \( {f}^{\prime }\left( x\right) = f\left( x\right) \) for all \( x \in \mathbf{R} \) . .4 Let \( R > 0 \), and let \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n},\mathop{\sum }\limits_{{n = 0}}^{\infty }{b}_{n}{x}^{n} \) be power series whose intervals of convergence include \( \left( {-R, R}\right) \) . Suppose that \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n} = \) \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{b}_{n}{x}^{n} \) for all \( x \in \left( {-R, R}\right) \) . Show that \( {a}_{n} = {b}_{n} \) for all \( n \) . .5 Prove that \( {\sin }^{\prime }x = \cos x \) and that \( {\cos }^{\prime }x = - \sin x \) for each \( x \in \mathbf{R} \) . (See Exercise (1.4.5:14) for the definition of sin and cos.) .6 Prove the trigonometric addition formulae: \[ \cos \left( {a + b}\right) = \cos a\cos b - \sin a\sin b \] \[ \sin \left( {a + b}\right) = \sin a\cos b + \cos a\sin b. \] (Define \[ F\left( x\right) = {\left( \cos \left( x + b\right) - \cos x\cos b + \sin x\sin b\right) }^{2} \] \[ + {\left( \sin \left( x + b\right) - \sin x\cos b - \cos x\sin b\right) }^{2} \] and consider \( {F}^{\prime }\left( x\right) \) .) .7 Prove that cos is a strictly decreasing function in the interval \( \left\lbrack {0,2}\right\rbrack \) , and that there is a unique \( p \) such that \( p/2 \in \left\lbrack {0,2}\right\rbrack \) and \( \cos \left( {p/2}\right) = 0 \) . Using the addition formulae from the preceding exercise, prove also that \( \cos \left( {x + {2p}}\right) = \cos x \) and \( \sin \left( {x + {2p}}\right) = \sin x \) for all \( x \in \mathbf{R} \) . (Of course, the number \( p \) is more usually denoted by \( \pi \) .) .8 Derive the binomial series \[ {\left( 1 + x\right) }^{\alpha } = \mathop{\sum }\limits_{{n = 0}}^{\infty }\left( \begin{array}{l} \alpha \\ n \end{array}\right) {x}^{n} \] for \( - 1 < x < 1 \), where \( \left( \begin{array}{l} \alpha \\ 0 \end{array}\right) = 1 \) and for \( n \geq 1 \) , \[ \left( \begin{array}{l} \alpha \\ n \end{array}\right) = \frac{\alpha \left( {\alpha - 1}\right) \left( {\alpha - 2}\right) \cdots \left( {\alpha - n + 1}\right) }{n!}. \] (First show that the series in question does converge for \( \left| x\right| < 1 \) . Then apply Taylor's Theorem with the Lagrange form of the remainder when \( 0 \leq x < 1 \), and with the Cauchy form when \( - 1 < x < 0 \) .) .9 Let \( p \geq 2,1/p + 1/q = 1 \), and \( 0 < c < 1 \) . Prove that \[ {\left( 1 + c\right) }^{q} + {\left( 1 - c\right) }^{q} - 2{\left( 1 + {c}^{p}\right) }^{q - 1} \geq 0. \] (Use the binomial series.) .10 Let \( x, y \in \mathbf{R} \), and let \( p, q \) be positive numbers with \( 1/p + 1/q = 1 \) . Use the preceding exercise to prove that \[ {\left| x + y\right| }^{q} + {\left| x - y\right| }^{q} \leq 2{\left( {\left| x\right| }^{p} + {\left| y\right| }^{p}\right) }^{q - 1} \] if \( 1 < p \leq 2 \), and that \[ {\left| x + y\right| }^{q} + {\left| x - y\right| }^{q} \geq 2{\left( {\left| x\right| }^{p} + {\left| y\right| }^{p}\right) }^{q - 1} \] if \( p \geq 2 \) . Although the Riemann integral is taught in elementary calculus courses, for the best part of a century, following the development of a more sophisticated integral by Lebesgue and others, it has had little practical value. Indeed, Dieudonné, in characteristically forthright mood ([13], page 142), claims that the Riemann integral "has at best the importance of a mildly interesting exercise in the general theory of measure and integration. Only the stubborn conservatism of academic tradition could freeze it into a regular part of the curriculum, long after it had outlived its historical importance." We believe, nevertheless, that it is worth presenting a rigorous development of the Riemann integral for both historical and paedagogical reasons; but we skip lightly over this material, leaving much of it to the exercises. We discuss the Lebesgue integral in Chapter 2. By a partition of a compact interval \( I = \left\lbrack {a, b}\right\rbrack \) we mean a finite sequence \( P = \left( {{x}_{0},{x}_{1},\ldots ,{x}_{n}}\right) \) of points of \( I \) such that \[ a = {x}_{0} \leq {x}_{1} \leq \cdots \leq {x}_{n} = b. \] The real number \[ \max \left\{ {{x}_{i + 1} - {x}_{i} : 0 \leq i \leq n - 1}\right\} \] is called the mesh of the partition. Loosely, we identify \( P \) with the set \( \left\{ {{x}_{0},\ldots ,{x}_{n}}\right\} \) . A partition \( Q \) is called a refinement of \( P \) if \( P \subset Q \) . Now let \( f : I \rightarrow \mathbf{R} \) a bounded function, and for \( 0 \leq i \leq n - 1 \) define \[ {m}_{i}\left( f\right) = \inf \left\{ {f\left( x\right) : {x}_{i} \leq x \leq {x}_{i + 1}}\right\} \] \[ {M}_{i}\left( f\right) = \sup \left\{ {f\left( x\right) : {x}_{i} \leq x \leq {x}_{i + 1}}\right\} . \] The real numbers \[ L\left( {f, P}\right) = \mathop{\sum }\limits_{{i = 0}}^{{n - 1}}{m}_{i}\left( f\right) \left( {{x}_{i + 1} - {x}_{i}}\right) , \] \[ U\left( {f, P}\right) = \mathop{\sum }\limits_{{i = 0}}^{{n - 1}}{M}_{i}\left( f\right) \left( {{x}_{i + 1} - {x}_{i}}\right) \] are called the lower sum and upper sum, respectively, for \( f \) and \( P \) . Since \[ \left( {b - a}\right) \inf f \leq L\left(

1008_(GTM174)Foundations of Real and Abstract Analysis

{x}_{i + 1} - {x}_{i} : 0 \leq i \leq n - 1}\right\} \] is called the mesh of the partition. Loosely, we identify \( P \) with the set \( \left\{ {{x}_{0},\ldots ,{x}_{n}}\right\} \) . A partition \( Q \) is called a refinement of \( P \) if \( P \subset Q \) . Now let \( f : I \rightarrow \mathbf{R} \) a bounded function, and for \( 0 \leq i \leq n - 1 \) define \[ {m}_{i}\left( f\right) = \inf \left\{ {f\left( x\right) : {x}_{i} \leq x \leq {x}_{i + 1}}\right\} \] \[ {M}_{i}\left( f\right) = \sup \left\{ {f\left( x\right) : {x}_{i} \leq x \leq {x}_{i + 1}}\right\} . \] The real numbers \[ L\left( {f, P}\right) = \mathop{\sum }\limits_{{i = 0}}^{{n - 1}}{m}_{i}\left( f\right) \left( {{x}_{i + 1} - {x}_{i}}\right) , \] \[ U\left( {f, P}\right) = \mathop{\sum }\limits_{{i = 0}}^{{n - 1}}{M}_{i}\left( f\right) \left( {{x}_{i + 1} - {x}_{i}}\right) \] are called the lower sum and upper sum, respectively, for \( f \) and \( P \) . Since \[ \left( {b - a}\right) \inf f \leq L\left( {f, P}\right) \leq U\left( {f, P}\right) \leq \left( {b - a}\right) \sup f \] the lower integral of \( f \) , \[ {\int }_{a}^{b}f = \sup \{ L\left( {f, P}\right) : P\text{ is a partition of }\left\lbrack {a, b}\right\rbrack \} , \] and the upper integral of \( f \) , \[ \overline{{\int }_{a}^{b}}f = \inf \{ U\left( {f, P}\right) : P\text{ is a partition of }\left\lbrack {a, b}\right\rbrack \} , \] exist. (1.5.9) Lemma. Let \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbf{R} \) be bounded, and let \( P, Q \) be partitions of \( \left\lbrack {a, b}\right\rbrack \) . Then \( L\left( {f, P}\right) \leq U\left( {f, Q}\right) \) . Proof. Take \( P = \left( {{x}_{0},\ldots ,{x}_{n}}\right) \), and first consider the case where \( Q = \) \( P \cup \{ \xi \} \) for some point \( \xi \notin P \) . Choose \( k \) such that \( {x}_{k} < \xi < {x}_{k + 1} \), and write \[ \alpha = \inf \left\{ {f\left( x\right) : {x}_{k} \leq x \leq \xi }\right\} \] \[ \beta = \inf \left\{ {f\left( x\right) : \xi \leq x \leq {x}_{k + 1}}\right\} . \] Then \( {m}_{k}\left( f\right) = \min \{ \alpha, b\} \), so \[ L\left( {f, P}\right) = \mathop{\sum }\limits_{\substack{{i = 0,} \\ {i \neq k} }}^{{n - 1}}{m}_{i}\left( f\right) \left( {{x}_{i + 1} - {x}_{i}}\right) + {m}_{k}\left( f\right) \left( {{x}_{k + 1} - {x}_{k}}\right) \] \[ = \mathop{\sum }\limits_{\substack{{i = 0,} \\ {i \neq k} }}^{{n - 1}}{m}_{i}\left( f\right) \left( {{x}_{i + 1} - {x}_{i}}\right) + {m}_{k}\left( f\right) \left( {\xi - {x}_{k}}\right) + {m}_{k}\left( f\right) \left( {{x}_{k + 1} - \xi }\right) \] \[ \leq \mathop{\sum }\limits_{\substack{{i = 0,} \\ {i \neq k} }}^{{n - 1}}{m}_{i}\left( f\right) \left( {{x}_{i + 1} - {x}_{i}}\right) + \alpha \left( {\xi - {x}_{k}}\right) + \beta \left( {{x}_{k + 1} - \xi }\right) \] \[ = L\left( {f, Q}\right) \text{.} \] Next, if \( Q \) is a refinement of \( P \), then \( Q = P \cup \left\{ {{\xi }_{1},\ldots ,{\xi }_{m}}\right\} \) for some distinct points \( {\xi }_{k} \notin P \), so \[ L\left( {f, P}\right) \leq L\left( {f, P \cup \left\{ {\xi }_{1}\right\} }\right) \] \[ \leq L\left( {f, P \cup \left\{ {{\xi }_{1},{\xi }_{2}}\right\} }\right) \] \[ \leq \cdots \] \[ \leq L\left( {f, Q}\right) \text{.} \] Similar arguments show that \( U\left( {f, Q}\right) \leq U\left( {f, P}\right) \) when \( Q \) is a refinement of \( P \) . Now consider any two partitions \( P, Q \) of \( \left\lbrack {a, b}\right\rbrack \) . Since \( P \cup Q \) is a refinement of both \( P \) and \( Q \), we have \[ L\left( {f, P}\right) \leq L\left( {f, P \cup Q}\right) \leq U\left( {f, P \cup Q}\right) \leq U\left( {f, Q}\right) . \] It follows from this lemma that, as we might have anticipated, \[ {\int }_{a}^{b}f \leq {\overline{\int }}_{a}^{b}f \] We say that \( f \) is Riemann integrable over \( I \) if its lower and upper integrals coincide, in which case we define the Riemann integral of \( f \) over \( I \) to be \[ {\int }_{a}^{b}f = {\int }_{a}^{b}f = \overline{{\int }_{a}^{b}}f. \] We also define \[ {\int }_{b}^{a}f = - {\int }_{a}^{b}f \] when \( f \) is Riemann integrable over \( \left\lbrack {a, b}\right\rbrack \) . ## (1.5.10) Exercises .1 Let \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbf{R} \) be bounded. Prove that \( f \) is Riemann integrable if and only if for each \( \varepsilon > 0 \) there exists a partition \( P \) of \( \left\lbrack {a, b}\right\rbrack \) such that \( U\left( {f, P}\right) - L\left( {f, P}\right) < \varepsilon \) . .2 Let \( f\left( x\right) = {x}^{2} \) on \( \left\lbrack {0,1}\right\rbrack \), and for each positive integer \( n \) let \( {P}_{n} \) be the partition of \( \left\lbrack {0,1}\right\rbrack \) consisting of the points \( i/n\left( {0 \leq i \leq n}\right) \) . By considering \( L\left( {f,{P}_{n}}\right) \) and \( U\left( {f,{P}_{n}}\right) \), show that \( f \) is Riemann integrable and that \( {\int }_{0}^{1}f = 1/3 \) . .3 Prove that for any \( n \in \mathbf{N} \) the function \( x \mapsto {x}^{n} \) is Riemann integrable over \( \left\lbrack {a, b}\right\rbrack \) . (Use the first exercise in this set.) .4 Prove that an increasing bounded function \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbf{R} \) is Riemann integrable. .5 Prove that a continuous function \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbf{R} \) is Riemann integrable. (Use the Uniform Continuity Theorem (Exercise (1.4.8:8)).) .6 Define \( f : \left\lbrack {0,1}\right\rbrack \rightarrow \mathbf{R} \) by \[ f\left( x\right) = \left\{ \begin{array}{ll} 1 & \text{ if }x\text{ is rational } \\ 0 & \text{ if }x\text{ is irrational. } \end{array}\right. \] Prove that \( f \) is not Riemann integrable. .7 Define \( f : \left\lbrack {0,1}\right\rbrack \rightarrow \mathbf{R} \) by \[ f\left( x\right) = \left\{ \begin{array}{ll} 0 & \text{ if }x = 0\text{ or }x\text{ is irrational } \\ & \\ \frac{1}{q} & \text{ if }x = \frac{p}{q}\text{ for relatively prime positive integers }p, q. \end{array}\right. \] Given \( \varepsilon > 0 \), let \( {t}_{0} = 0 \) ; let \( {t}_{1},\ldots ,{t}_{m - 1} \) be, in increasing order, the points of \( \left( {0,1}\right) \) that have the form \( p/q \) where \( p, q \) are relatively prime positive integers with \( 0 < q < 2/\varepsilon \) ; and let \( {t}_{m} = 1 \) . Taking \( {x}_{0} = 0 \) , construct inductively a partition \( P = \left( {{x}_{0},{x}_{1},\ldots ,{x}_{{2m} + 1}}\right) \) of \( \left\lbrack {0,1}\right\rbrack \) such that \[ {x}_{2k} < {t}_{k} < {x}_{{2k} + 1} < \frac{1}{2}\left( {{t}_{k} + {t}_{k + 1}}\right) \] \[ {x}_{{2k} + 1} - {x}_{2k} < \frac{\varepsilon }{2\left( {m + 1}\right) } \] and \( {x}_{{2m} + 1} = 1 \) . Show that if \( {x}_{{2k} - 1} \leq x \leq {x}_{2k} \), then \( f\left( x\right) < \varepsilon /2 \) ; that \( U\left( {f, P}\right) < \varepsilon \) ; and hence that \( f \) is Riemann integrable. .8 Prove that the composition of two Riemann integrable functions need not be Riemann integrable. (Note the preceding two exercises.) Since, by Exercise (1.5.10:3), \( f\left( x\right) = x \) defines a Riemann integrable function over \( \left\lbrack {a, b}\right\rbrack \), the next result generalises Exercise (1.5.10: 5). It should also be compared with Exercise (1.5.10: 8). (1.5.11) Proposition. Let \( f : \left\lbrack {a, b}\right\rbrack \rightarrow J \) be Riemann integrable, where \( J \) is a compact interval, and let \( g : J \rightarrow \mathbf{R} \) be continuous. Then \( g \circ f \) is Riemann integrable. Proof. Let \[ K = \sup \{ g\left( y\right) : y \in J\} \] which exists by Theorem (1.4.9), and write \( I = \left\lbrack {a, b}\right\rbrack \) . According to the Uniform Continuity Theorem (Exercise (1.4.8:8)), for each \( \varepsilon > 0 \) there exists \( \delta \) such that \[ 0 < \delta < \frac{\varepsilon }{b - a + {2K}} \] and such that \[ \left| {g\left( x\right) - g\left( y\right) }\right| < \frac{\varepsilon }{b - a + {2K}} \] whenever \( x, y \in J \) and \( \left| {x - y}\right| < \delta \) . Choose a partition \( P = \left( {{x}_{0},\ldots ,{x}_{n}}\right) \) of \( I \) such that \[ U\left( {f, P}\right) - L\left( {f, P}\right) < {\delta }^{2} \] (this is possible in view of Exercise (1.5.10:1)), and write \( P = S \cup T \), where \[ S = \left\{ {i : {M}_{i}\left( f\right) - {m}_{i}\left( f\right) < \delta }\right\} \] \[ T = \left\{ {i : {M}_{i}\left( f\right) - {m}_{i}\left( f\right) \geq \delta }\right\} . \] If \( i \in S \) and \( x,{x}^{\prime } \in \left\lbrack {{x}_{i},{x}_{i + 1}}\right\rbrack \), then \( \left| {f\left( x\right) - f\left( {x}^{\prime }\right) }\right| < \delta \) and so \[ \left| {g \circ f\left( x\right) - g \circ f\left( {x}^{\prime }\right) }\right| < \frac{\varepsilon }{b - a + {2K}}. \] Hence \[ {M}_{i}\left( {g \circ f}\right) - {m}_{i}\left( {g \circ f}\right) \leq \frac{\varepsilon }{b - a + {2K}} \] and therefore \[ \mathop{\sum }\limits_{{i \in S}}\left( {{M}_{i}\left( {g \circ f}\right) - {m}_{i}\left( {g \circ f}\right) }\right) \left( {{x}_{i + 1} - {x}_{i}}\right) \leq \frac{\varepsilon }{b - a + {2K}}\mathop{\sum }\limits_{{i = 0}}^{{n - 1}}\left( {{x}_{i + 1} - {x}_{i}}\right) \] \[ = \frac{\varepsilon \left( {b - a}\right) }{b - a + {2K}}. \] On the other hand, \[ \mathop{\sum }\limits_{{i \in T}}\left( {{x}_{i + 1} - {x}_{i}}\right) \leq {\delta }^{-1}\mathop{\sum }\limits_{{i \in T}}\left( {{M}_{i}\left( f\right) - {m}_{i}\left( f\right) }\right) \left( {{x}_{i + 1} - {x}_{i}}\right) \] \[ \leq {\delta }^{-1}\left( {U\left( {f, P}\right) - L\left( {f, P}\right) }\right) \] \[ < \delta \text{,} \] so \[ \mathop{\sum }\limits_{{i \in T}}\left( {{M}_{i}\left( {g \circ f}\right) - {m}_{i}\left( {g \circ f}\right) }\right) \left( {{x}_{i + 1} - {x}_{i}}\right) \leq {2K}\mathop{\sum }\limits_{{i \in T}}\left( {{x}_{i + 1} - {x}_{i}}\right) \] \[ < {2K\delta } \] \[ < \frac{2K\varepsilon }{b - a + {2K}}\tex

1008_(GTM174)Foundations of Real and Abstract Analysis

{{i \in S}}\left( {{M}_{i}\left( {g \circ f}\right) - {m}_{i}\left( {g \circ f}\right) }\right) \left( {{x}_{i + 1} - {x}_{i}}\right) \leq \frac{\varepsilon }{b - a + {2K}}\mathop{\sum }\limits_{{i = 0}}^{{n - 1}}\left( {{x}_{i + 1} - {x}_{i}}\right) \] \[ = \frac{\varepsilon \left( {b - a}\right) }{b - a + {2K}}. \] On the other hand, \[ \mathop{\sum }\limits_{{i \in T}}\left( {{x}_{i + 1} - {x}_{i}}\right) \leq {\delta }^{-1}\mathop{\sum }\limits_{{i \in T}}\left( {{M}_{i}\left( f\right) - {m}_{i}\left( f\right) }\right) \left( {{x}_{i + 1} - {x}_{i}}\right) \] \[ \leq {\delta }^{-1}\left( {U\left( {f, P}\right) - L\left( {f, P}\right) }\right) \] \[ < \delta \text{,} \] so \[ \mathop{\sum }\limits_{{i \in T}}\left( {{M}_{i}\left( {g \circ f}\right) - {m}_{i}\left( {g \circ f}\right) }\right) \left( {{x}_{i + 1} - {x}_{i}}\right) \leq {2K}\mathop{\sum }\limits_{{i \in T}}\left( {{x}_{i + 1} - {x}_{i}}\right) \] \[ < {2K\delta } \] \[ < \frac{2K\varepsilon }{b - a + {2K}}\text{.} \] It now follows that \[ U\left( {g \circ f, P}\right) - L\left( {g \circ f, P}\right) \] \[ = \mathop{\sum }\limits_{{i \in S}}\left( {{M}_{i}\left( {g \circ f}\right) - {m}_{i}\left( {g \circ f}\right) }\right) \left( {{x}_{i + 1} - {x}_{i}}\right) \] \[ + \mathop{\sum }\limits_{{i \in T}}\left( {{M}_{i}\left( {g \circ f}\right) - {m}_{i}\left( {g \circ f}\right) }\right) \left( {{x}_{i + 1} - {x}_{i}}\right) \] \[ < \frac{\varepsilon \left( {b - a}\right) }{b - a + {2K}} + \frac{2K\varepsilon }{b - a + {2K}} = \varepsilon . \] Reference to Exercise (1.5.10:1) completes the proof. Let \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbf{R} \) be a bounded function, and \( P = \left( {{x}_{0},\ldots ,{x}_{n}}\right) \) a partition of \( \left\lbrack {a, b}\right\rbrack \) . Any expression of the form \[ \mathop{\sum }\limits_{{i = 0}}^{{n - 1}}f\left( {\xi }_{i}\right) \left( {{x}_{i + 1} - {x}_{i}}\right) \] where \( {\xi }_{i} \in \left\lbrack {{x}_{i},{x}_{i + 1}}\right\rbrack \) for each \( i \), is called a Riemann sum for \( f \) (relative to \( P) \) . If \( f \) is Riemann integrable, geometric arguments like those presented in elementary calculus courses lead us to believe that if the partition \( P \) has small mesh, then the corresponding Riemann sums will closely approximate \( {\int }_{a}^{b}f \) . This expectation is fulfilled in the first of the next set of exercises. ## (1.5.12) Exercises . 1 Prove that a bounded function \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbf{R} \) is Riemann integrable if and only if there exists a real number \( \Lambda \) with the following property. For each \( \varepsilon > 0 \) there exists \( \delta > 0 \) such that \( \left| {\sum - \Lambda }\right| < \varepsilon \) whenever - \( P \) is a partition of \( \left\lbrack {a, b}\right\rbrack \) with mesh less than \( \delta \), and - \( \sum \) is a Riemann sum for \( f \) relative to \( P \) . Prove that, in that case, \( \Lambda = {\int }_{a}^{b}f \) . .2 Let \( f, g \) be Riemann integrable functions on \( \left\lbrack {a, b}\right\rbrack \), and let \( \lambda \in \mathbf{R} \) . Prove that \( f + g \) and \( {\lambda f} \) are Riemann integrable, and that \[ {\int }_{a}^{b}\left( {f + g}\right) = {\int }_{a}^{b}f + {\int }_{a}^{b}g \] \[ {\int }_{a}^{b}\left( {\lambda f}\right) = \lambda {\int }_{a}^{b}f. \] (Use the preceding exercise.) .3 Let \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbf{R} \) be bounded, and \( a \leq c \leq b \) . Prove that \( f \) is Riemann integrable over \( \left\lbrack {a, b}\right\rbrack \) if and only if it is Riemann integrable over both \( \left\lbrack {a, c}\right\rbrack \) and \( \left\lbrack {c, b}\right\rbrack \), in which case \[ {\int }_{a}^{b}f = {\int }_{a}^{c}f + {\int }_{c}^{b}f \] .4 Let \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbf{R} \) be Riemann integrable. Prove that \( {f}^{ + } = f \vee 0 \) , \( {f}^{ - } = \left( {-f}\right) \vee 0 \), and \( \left| f\right| \) are Riemann integrable, and that \( \left| {{\int }_{a}^{b}f}\right| \leq \) \( {\int }_{a}^{b}\left| f\right| \) . Prove also that if \( \left| {f\left( x\right) }\right| \leq M \) for all \( x \in \left\lbrack {a, b}\right\rbrack \), then \[ \left| {{\int }_{x}^{y}f}\right| \leq M\left| {x - y}\right| \] for all \( x, y \in \left\lbrack {a, b}\right\rbrack \) . .5 Let \( f \) and \( g \) be Riemann integrable functions on \( \left\lbrack {a, b}\right\rbrack \) . Give two proofs that the product function \( {fg} \) is Riemann integrable over \( \left\lbrack {a, b}\right\rbrack \) . (For one proof, first take \( f \geq 0 \) and use Exercise (1.5.10:1). For a second proof, note that \( {fg} = \frac{1}{4}\left( {{\left( f + g\right) }^{2} - {\left( f - g\right) }^{2}}\right) \) .) .6 Let \( f \) be a nonvanishing Riemann integrable function on the compact interval \( I = \left\lbrack {a, b}\right\rbrack \), and suppose that \( 1/f \) is bounded on \( I \) . Show that \( 1/f \) is Riemann integrable over \( I \) . .7 Prove that if \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbf{R} \) is continuous and nonnegative, and \( {\int }_{a}^{b}f = 0 \), then \( f\left( x\right) = 0 \) for all \( x \in \left\lbrack {a, b}\right\rbrack \) . What makes the calculation of integrals feasible is the connection between integration and differentiation. There are various expressions of this connection, each of which may lay claim to the historic title of Fundamental Theorem of Calculus. Here is one strong version of that theorem. (1.5.13) Theorem. If \( F \) is differentiable on \( \left\lbrack {a, b}\right\rbrack \), and \( {F}^{\prime } \) is Riemann integrable over \( \left\lbrack {a, b}\right\rbrack \), then \[ {\int }_{a}^{b}{F}^{\prime } = F\left( b\right) - F\left( a\right) \] Proof. Let \( P = \left( {{x}_{0},\ldots ,{x}_{n}}\right) \) be any partition of \( \left\lbrack {a, b}\right\rbrack \) . By the Mean Value Theorem, for each \( i \) there exists \( {\xi }_{i} \in \left( {{x}_{i},{x}_{i + 1}}\right) \) such that \[ F\left( {x}_{i + 1}\right) - F\left( {x}_{i}\right) = {F}^{\prime }\left( {\xi }_{i}\right) \left( {{x}_{i + 1} - {x}_{i}}\right) . \] Hence \[ \mathop{\sum }\limits_{{i = 0}}^{{n - 1}}{F}^{\prime }\left( {\xi }_{i}\right) \left( {{x}_{i + 1} - {x}_{i}}\right) = \mathop{\sum }\limits_{{i = 0}}^{{n - 1}}\left( {F\left( {x}_{i + 1}\right) - F\left( {x}_{i}\right) }\right) = F\left( b\right) - F\left( a\right) . \] Since \( P \) is any partition of \( \left\lbrack {a, b}\right\rbrack \), the result now follows from Exercise \( \left( {{1.5.12} : 1}\right) \) . Let \( F, f \) be two mappings of \( \left\lbrack {a, b}\right\rbrack \) into \( \mathbf{R} \) . We say that \( F \) is a primitive, or antiderivative, of \( f \) on \( \left\lbrack {a, b}\right\rbrack \) if \( {F}^{\prime }\left( x\right) = f\left( x\right) \) for all \( x \in \left\lbrack {a, b}\right\rbrack \) . In view of the Fundamental Theorem of Calculus (1.5.13), there is an obvious strategy for calculating the Riemann integral of a function \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbf{R} \) : first find a primitive \( F \) of \( f \), and then compute \( F\left( b\right) - F\left( a\right) \) . Of course, as any student of calculus quickly learns, finding primitives of Riemann integrable functions is often a (literally) tricky business; moreover, the class of Riemann integrable functions \( f \) for which there exist primitives expressible in terms of elementary functions is relatively small [41]. So the Fundamental Theorem of Calculus has severe practical limitations, which have led to the development of highly accurate, fast methods of numerical integration (see, for example, [26]). ## (1.5.14) Exercises .1 Let \( f \) be Riemann integrable over \( I = \left\lbrack {a, b}\right\rbrack \), and define \[ F\left( x\right) = {\int }_{a}^{x}f\;\left( {a \leq x \leq b}\right) . \] Prove that \( F \) is continuous on \( I \) . Prove also that if \( {x}_{0} \in I \) and \[ \mathop{\lim }\limits_{{x \rightarrow {x}_{0}, x \in I}}f\left( x\right) = f\left( {x}_{0}\right) \] then \[ \mathop{\lim }\limits_{{x \rightarrow {x}_{0}, x \in I}}\frac{F\left( x\right) - F\left( {x}_{0}\right) }{x - {x}_{0}} = f\left( {x}_{0}\right) . \] In particular, we obtain a result which is also sometimes called the Fundamental Theorem of Calculus: if \( f \) is continuous on \( \left\lbrack {a, b}\right\rbrack \), then \( F \) is differentiable on \( \left\lbrack {a, b}\right\rbrack ,{F}^{\prime }\left( x\right) = f\left( x\right) \) for all \( x \in \left( {a, b}\right) ,{F}^{\prime }\left( {a}^{ + }\right) = \) \( f\left( a\right) \), and \( {F}^{\prime }\left( {b}^{ - }\right) = f\left( b\right) \) . .2 Let the power series \( f\left( x\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{x}^{n} \) have radius of convergence \( R \) (which could be \( \infty \) ). Prove that for each \( x \in \left( {-R, R}\right) \) , \[ {\int }_{0}^{x}f = \mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{{a}_{n}}{n + 1}{x}^{n + 1}. \] .3 By considering \[ \left( {1 + x}\right) \left( {1 - x + {x}^{2} - {x}^{3} + \cdots + {\left( -1\right) }^{n}{x}^{n}}\right) , \] show that \[ \frac{1}{1 + x} = \mathop{\sum }\limits_{{n = 0}}^{\infty }{\left( -1\right) }^{n}{x}^{n}\;\left( {-1 < x < 1}\right) . \] Hence find power series expansions for \( \log \left( {1 - x}\right) \) and \( {\left( 1 - x\right) }^{-1}\log (1 - \) \( x) \) on \( \left( {-1,1}\right) \) . Then show that the identity \[ \frac{1}{2}{\left( \log \left( 1 - x\right) \right) }^{2} = \frac{1}{2}{x}^{2} + \frac{1}{3}\left( {1 + \frac{1}{2}}\right) {x}^{3} + \frac{1}{4}\left( {1 + \frac{1}{2} + \frac{1}{3}}\right) {x}^{4} + \cdots \] is valid for each \( x \in ( - 1,1\rbrack \) . .4 In this exercise you will prove that \( {\pi }^{2} \), and therefore \( \pi \) itself, is irrational. Given a positive i

1008_(GTM174)Foundations of Real and Abstract Analysis

that for each \( x \in \left( {-R, R}\right) \) , \[ {\int }_{0}^{x}f = \mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{{a}_{n}}{n + 1}{x}^{n + 1}. \] .3 By considering \[ \left( {1 + x}\right) \left( {1 - x + {x}^{2} - {x}^{3} + \cdots + {\left( -1\right) }^{n}{x}^{n}}\right) , \] show that \[ \frac{1}{1 + x} = \mathop{\sum }\limits_{{n = 0}}^{\infty }{\left( -1\right) }^{n}{x}^{n}\;\left( {-1 < x < 1}\right) . \] Hence find power series expansions for \( \log \left( {1 - x}\right) \) and \( {\left( 1 - x\right) }^{-1}\log (1 - \) \( x) \) on \( \left( {-1,1}\right) \) . Then show that the identity \[ \frac{1}{2}{\left( \log \left( 1 - x\right) \right) }^{2} = \frac{1}{2}{x}^{2} + \frac{1}{3}\left( {1 + \frac{1}{2}}\right) {x}^{3} + \frac{1}{4}\left( {1 + \frac{1}{2} + \frac{1}{3}}\right) {x}^{4} + \cdots \] is valid for each \( x \in ( - 1,1\rbrack \) . .4 In this exercise you will prove that \( {\pi }^{2} \), and therefore \( \pi \) itself, is irrational. Given a positive integer \( n \), define \[ \phi \left( x\right) = \frac{1}{n!}{x}^{n}{\left( 1 - x\right) }^{n}. \] Prove that (i) \( {\phi }^{\left( k\right) }\left( 0\right) = 0 \) for \( k < n \) or \( k > {2n} \) ; (ii) \( {\phi }^{\left( k\right) }\left( 0\right) \) and \( {\phi }^{\left( k\right) }\left( 1\right) \) are integers for all \( k \in \mathbf{N} \) . Suppose that \( {\pi }^{2} = p/q \) for some positive integers \( p, q \), and define \[ F = {q}^{n}\mathop{\sum }\limits_{{k = 0}}^{n}{\left( -1\right) }^{{2n} - k}{\pi }^{2\left( {n - k}\right) }{\phi }^{\left( 2k\right) }. \] Show that \( F\left( 0\right) \) and \( F\left( 1\right) \) are integers, and that \[ {F}^{\prime \prime }\left( x\right) + {\pi }^{2}F\left( x\right) = {\pi }^{2}{p}^{n}\phi \left( x\right) . \] Setting \[ G\left( x\right) = {F}^{\prime }\left( x\right) \sin {\pi x} - {\pi F}\left( x\right) \cos {\pi x}, \] show that \[ {G}^{\prime }\left( x\right) = {\pi }^{2}{p}^{n}\phi \left( x\right) \sin {\pi x} \] and hence that \[ \pi {\int }_{0}^{1}{p}^{n}\phi \left( x\right) \sin {\pi x}\mathrm{\;d}x \] is an integer. Finally, show that \[ 0 < \pi {\int }_{0}^{1}{p}^{n}\phi \left( x\right) \sin {\pi x}\mathrm{\;d}x < \frac{\pi {p}^{n}}{n!} \] for all positive integers \( n \), and derive a contradiction. We end this chapter by sketching the development of a generalisation of the Riemann integral. To do so we must first introduce another important class of functions on intervals. In the rest of the section, unless we say otherwise, \( I = \left\lbrack {a, b}\right\rbrack \) is a compact interval, and \( f \) a mapping of \( I \) into \( \mathbf{R} \) . For all \( x, y \in I \) with \( x \leq y \) we define the variation of \( f \) over \( \left\lbrack {x, y}\right\rbrack \) to be \[ {T}_{f}\left( {x, y}\right) = \sup \left\{ {\mathop{\sum }\limits_{{i = 0}}^{{n - 1}}\left| {f\left( {x}_{i + 1}\right) - f\left( {x}_{i}\right) }\right| : x = {x}_{0} \leq {x}_{1} \leq \cdots \leq {x}_{n} = y}\right\} \] if this quantity exists as a real number; we then say that \( f \) has bounded variation on \( \left\lbrack {x, y}\right\rbrack \) . ## (1.5.15) Exercises .1 Let \( f \) have bounded variation on \( I \) . Prove that (i) \( f \) is bounded on \( I \) , (ii) \( {T}_{f}\left( {a, b}\right) = {T}_{f}\left( {a, x}\right) + {T}_{f}\left( {x, b}\right) \) for all \( x \in I \), and (iii) \( {T}_{f}\left( {a, \cdot }\right) \) is an increasing function on \( I \) . .2 Prove that \( f \) has bounded variation on \( I \) if and only if there exist increasing functions \( g, h \) on \( I \) such that \( f = g - h \) . (For "only if" note part (iii) of the preceding exercise.) .3 Let \( f, g \) be functions of bounded variation on \( I \) . Prove that \( f + g \) , \( {\lambda f} \) (where \( \lambda \in \mathbf{R} \) ), and \( {fg} \) are of bounded variation, and that if \( \inf \{ \left| {f\left( x\right) }\right| : x \in I\} > 0 \), then \( 1/f \) is of bounded variation. .4 Define \( f : \left\lbrack {0,1}\right\rbrack \rightarrow \mathbf{R} \) by \[ f\left( x\right) = \left\{ \begin{array}{ll} {x}^{2}\sin \frac{1}{{x}^{2}} & \text{ if }0 < x \leq 1 \\ & \\ 0 & \text{ if }x = 0. \end{array}\right. \] Prove that \( f \) is differentiable at each point of \( \left\lbrack {0,1}\right\rbrack \) but does not have bounded variation on \( \left\lbrack {0,1}\right\rbrack \) . .5 Let \( f : I \rightarrow \mathbf{R} \) have bounded variation. Prove that the one-sided limits \( f\left( {x}^{ - }\right) \) and \( f\left( {x}^{ + }\right) \) exist at each point of \( \left( {a, b}\right) \), as do \( f\left( {a}^{ + }\right) \) and \( f\left( {b}^{ - }\right) \), and that the set of points of \( I \) at which \( f \) is discontinuous is either empty or countable. (See Exercise (1.4.5:8).) .6 Let \( \left( {f}_{n}\right) \) be a sequence of functions of bounded variation on \( I \) such that \( f\left( x\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}{f}_{n}\left( x\right) \) exists for each \( x \in I \) . Prove that \( {T}_{f}\left( {a, b}\right) \leq \) \( \liminf {T}_{{f}_{n}}\left( {a, b}\right) \) . (First show that for any partition \( \left( {{x}_{0},{x}_{1},\ldots ,{x}_{n}}\right) \) of \( I \) and any positive integer \( k \) , \[ \mathop{\sum }\limits_{{i = 0}}^{{n - 1}}\left| {f\left( {x}_{i + 1}\right) - f\left( {x}_{i}\right) }\right| \leq {T}_{{f}_{k}}\left( {a, b}\right) + \mathop{\sum }\limits_{{i = 0}}^{{n - 1}}\left| {f\left( {x}_{i + 1}\right) - {f}_{k}\left( {x}_{i + 1}\right) }\right| \] \[ + \mathop{\sum }\limits_{{i = 0}}^{{n - 1}}\left| {f\left( {x}_{i}\right) - {f}_{k}\left( {x}_{i}\right) }\right| . \] Given \( \varepsilon > 0 \), then choose \( k \) appropriately.) Now let \( \alpha : I \rightarrow \mathbf{R} \) be a function with bounded variation on \( I, P = \) \( \left( {{x}_{0},{x}_{1},\ldots ,{x}_{n}}\right) \) a partition of \( I \), and \( f : I \rightarrow \mathbf{R} \) a bounded function. Any expression of the form \[ \mathop{\sum }\limits_{{i = 0}}^{{n - 1}}f\left( {\xi }_{i}\right) \left( {\alpha \left( {x}_{i + 1}\right) - \alpha \left( {x}_{i}\right) }\right) \] where \( {\xi }_{i} \in \left\lbrack {{x}_{i},{x}_{i + 1}}\right\rbrack \) for each \( i \), is called a Riemann-Stieltjes sum for \( f \) (relative to \( P \) and \( \alpha \) ). We say that \( f \) is Riemann-Stieltjes integrable (over \( I \) with respect to \( \alpha \) ) if there exists a real number \( \Lambda \) with the following property. For each \( \varepsilon > 0 \) there exists \( \delta > 0 \) such that \( \left| {\sum - \Lambda }\right| < \varepsilon \) whenever - \( P \) is a partition of \( I \) with mesh less than \( \delta \) and - \( \sum \) is a Riemann-Stieltjes sum for \( f \) relative to \( P \) . In that case, \( \Lambda \) -the Riemann-Stieltjes integral of \( f \) with respect to \( \alpha \) -is the unique real number with this property, and is usually written \[ {\int }_{a}^{b}f\left( x\right) \mathrm{d}\alpha \left( x\right) \] Exercise (1.5.12:1) shows that the Riemann integral is just the special case of the Riemann-Stieltjes integral in which \( \alpha \left( x\right) = x \) . ## (1.5.16) Exercises In these exercises, \( \alpha \) has bounded variation on \( I \), and \( f, g \) are bounded real-valued functions on \( I \) . .1 Why, in the foregoing definitions, do we require the function \( \alpha \) to be of bounded variation? .2 Let \( \alpha \) be an increasing function on \( I = \left\lbrack {a, b}\right\rbrack \) . Given a partition \( P = \) \( \left( {{x}_{0},{x}_{1},\ldots ,{x}_{n}}\right) \) of \( I \), we call the real numbers \[ L\left( {f, P,\alpha }\right) = \mathop{\sum }\limits_{{i = 0}}^{{n - 1}}{m}_{i}\left( f\right) \left( {\alpha \left( {x}_{i + 1}\right) - \alpha \left( {x}_{i}\right) }\right) , \] \[ U\left( {f, P,\alpha }\right) = \mathop{\sum }\limits_{{i = 0}}^{{n - 1}}{M}_{i}\left( f\right) \left( {\alpha \left( {x}_{i + 1}\right) - \alpha \left( {x}_{i}\right) }\right) , \] respectively, the lower sum and the upper sum for \( f \) relative to \( P \) and \( \alpha \) . Prove that the lower integral \[ {\int }_{a}^{b}f\left( x\right) \mathrm{d}\alpha \left( x\right) = \sup \{ L\left( {f, P,\alpha }\right) : P\text{ is a partition of }I\} \] and the upper integral \[ \overline{{\int }_{a}^{b}}f\left( x\right) \mathrm{d}\alpha \left( x\right) = \inf \{ U\left( {f, P,\alpha }\right) : P\text{ is a partition of }I\} \] of \( f \) with respect to \( \alpha \) exist. Prove also that \( f \) is Riemann-Stieltjes integrable with respect to \( \alpha \) if and only if \[ {\int }_{a}^{b}f\left( x\right) \mathrm{d}\alpha \left( x\right) = \overline{{\int }_{a}^{b}}f\left( x\right) \mathrm{d}\alpha \left( x\right) \] in which case their common value is \( {\int }_{a}^{b}f\left( x\right) \mathrm{d}\alpha \left( x\right) \) . .3 Prove that if \( f \) is continuous, then it is Riemann-Stieltjes integrable with respect to \( \alpha \), and \[ \left| {{\int }_{a}^{b}f\left( x\right) \mathrm{d}\alpha \left( x\right) }\right| \leq {\int }_{a}^{b}\left| {f\left( x\right) }\right| \mathrm{d}{T}_{\alpha }\left( {a, x}\right) \leq M{T}_{\alpha }\left( {a, b}\right) \] where \( {T}_{\alpha }\left( {a, x}\right) \) is the variation of \( \alpha \) on the interval \( \left\lbrack {a, x}\right\rbrack \), and \( M = \) \( \sup \{ \left| {f\left( x\right) }\right| : a \leq x \leq b\} \) . (Note that \( {T}_{\alpha }\left( {a, \cdot }\right) \) has bounded variation, by Exercises (1.5.15: 1 and 2).) .4 Prove that if \( f, g \) are Riemann-Stieltjes integrable with respect to \( \alpha \) , then so are \( f + g, f - g \), and \( {\lambda f} \) (where \( \lambda \in \mathbf{R} \) ); in which case, \[ {\int }_{a}^{b}\left( {f\left( x\right) + g\left( x\right) }\right) \mathrm{d}\alpha \left( x\right) = {\int }_{a}^{b}f\left( x\right) \mathrm{d}\alpha \left( x\right) + {\int }_{a}^{b}g\left( x\r

1008_(GTM174)Foundations of Real and Abstract Analysis

rable with respect to \( \alpha \), and \[ \left| {{\int }_{a}^{b}f\left( x\right) \mathrm{d}\alpha \left( x\right) }\right| \leq {\int }_{a}^{b}\left| {f\left( x\right) }\right| \mathrm{d}{T}_{\alpha }\left( {a, x}\right) \leq M{T}_{\alpha }\left( {a, b}\right) \] where \( {T}_{\alpha }\left( {a, x}\right) \) is the variation of \( \alpha \) on the interval \( \left\lbrack {a, x}\right\rbrack \), and \( M = \) \( \sup \{ \left| {f\left( x\right) }\right| : a \leq x \leq b\} \) . (Note that \( {T}_{\alpha }\left( {a, \cdot }\right) \) has bounded variation, by Exercises (1.5.15: 1 and 2).) .4 Prove that if \( f, g \) are Riemann-Stieltjes integrable with respect to \( \alpha \) , then so are \( f + g, f - g \), and \( {\lambda f} \) (where \( \lambda \in \mathbf{R} \) ); in which case, \[ {\int }_{a}^{b}\left( {f\left( x\right) + g\left( x\right) }\right) \mathrm{d}\alpha \left( x\right) = {\int }_{a}^{b}f\left( x\right) \mathrm{d}\alpha \left( x\right) + {\int }_{a}^{b}g\left( x\right) \mathrm{d}\alpha \left( x\right) , \] \[ {\int }_{a}^{b}\left( {f\left( x\right) - g\left( x\right) }\right) \mathrm{d}\alpha \left( x\right) = {\int }_{a}^{b}f\left( x\right) \mathrm{d}\alpha \left( x\right) - {\int }_{a}^{b}g\left( x\right) \mathrm{d}\alpha \left( x\right) , \] and \[ {\int }_{a}^{b}{\lambda f}\left( x\right) \mathrm{d}\alpha \left( x\right) = \lambda {\int }_{a}^{b}f\left( x\right) \mathrm{d}\alpha \left( x\right) . \] .5 Let \( a \leq c \leq b \) . Prove that \( f \) is Riemann-Stieltjes integrable over \( \left\lbrack {a, b}\right\rbrack \) (with respect to \( \alpha \) ) if and only if it is Riemann-Stieltjes integrable over both \( \left\lbrack {a, c}\right\rbrack \) and \( \left\lbrack {c, b}\right\rbrack \) ; in which case, \[ {\int }_{a}^{b}f\left( x\right) \mathrm{d}\alpha \left( x\right) = {\int }_{a}^{c}f\left( x\right) \mathrm{d}\alpha \left( x\right) + {\int }_{c}^{b}f\left( x\right) \mathrm{d}\alpha \left( x\right) . \] .6 Prove that if \( \alpha \) has a continuous derivative on \( I \), then the Riemann-Stieltjes integral \( {\int }_{a}^{b}f\left( x\right) \mathrm{d}\alpha \left( x\right) \) exists and equals the Riemann integral \( {\int }_{a}^{b}f{\alpha }^{\prime } \) .7 Let \( \alpha ,\beta \) be of bounded variation on \( I \), and suppose that \( f \) is Riemann-Stieltjes integrable with respect to both \( \alpha \) and \( \beta \) . Prove that \( f \) is Riemann-Stieltjes integrable with respect to \( \alpha + \beta \), and that \[ {\int }_{a}^{b}f\left( x\right) \mathrm{d}\left( {\alpha + \beta }\right) \left( x\right) = {\int }_{a}^{b}f\left( x\right) \mathrm{d}\alpha \left( x\right) + {\int }_{a}^{b}f\left( x\right) \mathrm{d}\beta \left( x\right) . \] Prove also that for each \( \lambda \in \mathbf{R}, f \) is Riemann-Stieltjes integrable with respect to \( {\lambda \alpha } \), and \[ {\int }_{a}^{b}f\left( x\right) \mathrm{d}\left( {\lambda \alpha }\right) \left( x\right) = \lambda {\int }_{a}^{b}f\left( x\right) \mathrm{d}\alpha \left( x\right) . \] The next lemma enables us to discuss the continuity of the function \( {T}_{f}\left( {a, \cdot }\right) \) . (1.5.17) Lemma. Let \( f \) have bounded variation on \( I \) . Then for each positive integer \( n \) there exists a function \( {g}_{n} : I \rightarrow \mathbf{R} \) such that (i) \( \left| {f\left( {x}^{ + }\right) - f\left( x\right) }\right| = \left| {{g}_{n}\left( {x}^{ + }\right) - {g}_{n}\left( x\right) }\right| \) whenever \( a \leq x < b \) , (ii) \( \left| {f\left( x\right) - f\left( {x}^{ - }\right) }\right| = \left| {{g}_{n}\left( x\right) - {g}_{n}\left( {x}^{ - }\right) }\right| \) whenever \( a < x \leq b \) , (iii) \( {T}_{f}\left( {a, \cdot }\right) - {g}_{n} \) is an increasing function, and (iv) \( 0 \leq {T}_{f}\left( {a, x}\right) - {g}_{n}\left( x\right) < 1/n \) for all \( x \in I \) . Moreover, \( f \) and \( {g}_{n} \) are continuous at precisely the same points of \( I \) . Proof. We may assume that \( a < b \) . Referring to Exercise (1.5.15:5), choose points \( a = {x}_{0} < {x}_{1} < \cdots < {x}_{m - 1} < {x}_{m} = b \) such that \( f \) is continuous at \( {x}_{i}\left( {1 \leq i \leq m - 1}\right) \), and \[ \mathop{\sum }\limits_{{i = 0}}^{{m - 1}}\left| {f\left( {x}_{i + 1}\right) - f\left( {x}_{i}\right) }\right| > {T}_{f}\left( {a, b}\right) - \frac{1}{n}. \] Setting \( {g}_{n}\left( a\right) = 0 \), construct the function \( {g}_{n} \) on the intervals \( \left\lbrack {{x}_{i},{x}_{i + 1}}\right\rbrack \) inductively, as follows. Assume that \( {g}_{n}\left( x\right) \) has been defined for \( a \leq x \leq {x}_{i} \) , where \( i < m \), and consider \( x \) with \( {x}_{i} < x \leq {x}_{i + 1} \) . If \( f\left( {x}_{i + 1}\right) - f\left( {x}_{i}\right) \geq 0 \) , set \[ {g}_{n}\left( x\right) = f\left( x\right) + {g}_{n}\left( {x}_{i}\right) - f\left( {x}_{i}\right) \] if \( f\left( {x}_{i + 1}\right) - f\left( {x}_{i}\right) < 0 \), set \[ {g}_{n}\left( x\right) = - f\left( x\right) + {g}_{n}\left( {x}_{i}\right) + f\left( {x}_{i}\right) . \] This completes the inductive construction. If \( {x}_{i} \leq x < {x}^{\prime } \leq {x}_{i + 1} \), then (see Exercise (1.5.15:1)) \[ {T}_{f}\left( {a,{x}^{\prime }}\right) - {T}_{f}\left( {a, x}\right) = {T}_{f}\left( {x,{x}^{\prime }}\right) \] \[ \geq \left| {f\left( {x}^{\prime }\right) - f\left( x\right) }\right| \] \[ = \left| {{g}_{n}\left( {x}^{\prime }\right) - {g}_{n}\left( x\right) }\right| \] \[ \geq {g}_{n}\left( {x}^{\prime }\right) - {g}_{n}\left( x\right) \] Thus if \( {x}_{i - 1} \leq x < {x}_{i} < \cdots < {x}_{j} < {x}^{\prime } \leq {x}_{j + 1} \), then \[ {T}_{f}\left( {a,{x}^{\prime }}\right) - {T}_{f}\left( {a, x}\right) = {T}_{f}\left( {a,{x}^{\prime }}\right) - {T}_{f}\left( {a,{x}_{j}}\right) \] \[ + \mathop{\sum }\limits_{{k = i}}^{{j - 1}}\left( {{T}_{f}\left( {a,{x}_{k + 1}}\right) - {T}_{f}\left( {a,{x}_{k}}\right) }\right) \] \[ + {T}_{f}\left( {a,{x}_{i}}\right) - {T}_{f}\left( {a, x}\right) \] \[ \geq {g}_{n}\left( {x}^{\prime }\right) - {g}_{n}\left( {x}_{j}\right) + \mathop{\sum }\limits_{{k = i}}^{{j - 1}}\left( {{g}_{n}\left( {x}_{k + 1}\right) - {g}_{n}\left( {x}_{k}\right) }\right) \] \[ + {g}_{n}\left( {x}_{i}\right) - {g}_{n}\left( x\right) \] \[ = {g}_{n}\left( {x}^{\prime }\right) - {g}_{n}\left( x\right) . \] It follows that \( {T}_{f}\left( {a, \cdot }\right) - {g}_{n} \) is an increasing function. For all \( x \in \left\lbrack {a, b}\right\rbrack \) we now have \[ 0 = {T}_{f}\left( {a, a}\right) - {g}_{n}\left( a\right) \] \[ \leq {T}_{f}\left( {a, x}\right) - {g}_{n}\left( x\right) \] \[ \leq {T}_{f}\left( {a, b}\right) - {g}_{n}\left( b\right) \] \[ = {T}_{f}\left( {a, b}\right) - \mathop{\sum }\limits_{{i = 0}}^{{n - 1}}\left( {{g}_{n}\left( {x}_{i + 1}\right) - {g}_{n}\left( {x}_{i}\right) }\right) \] \[ = {T}_{f}\left( {a, b}\right) - \mathop{\sum }\limits_{{i = 0}}^{{n - 1}}\left| {f\left( {x}_{i + 1}\right) - f\left( {x}_{i}\right) }\right| \] \[ < \frac{1}{n}\text{.}\] Finally, properties (i) and (ii) hold, since on each interval \( \left\lbrack {{x}_{i},{x}_{i + 1}}\right\rbrack \) either \( {g}_{n} - f \) or \( {g}_{n} + f \) is constant; the last part of the statement of the lemma follows immediately. (1.5.18) Proposition. Let \( f \) have bounded variation on \( I \) . Then \[ {T}_{f}\left( {a,{x}^{ + }}\right) - {T}_{f}\left( {a, x}\right) = \left| {f\left( {x}^{ + }\right) - f\left( x\right) }\right| \] if \( a \leq x < b \), and \[ {T}_{f}\left( {a, x}\right) - {T}_{f}\left( {a,{x}^{ - }}\right) = \left| {f\left( x\right) - f\left( {x}^{ - }\right) }\right| \] if \( a < x \leq b \) . Hence \( f \) and \( {T}_{f}\left( {a, \cdot }\right) \) are continuous at precisely the same points of \( I \) . Proof. For each positive integer \( n \) choose \( {g}_{n} \) as in the preceding lemma. For \( a \leq x < {x}^{\prime } < b \) we have \[ \left| {{T}_{f}\left( {a,{x}^{\prime }}\right) - {T}_{f}\left( {a, x}\right) - {g}_{n}\left( {x}^{\prime }\right) + {g}_{n}\left( x\right) }\right| \] \[ \leq \left| {{T}_{f}\left( {a,{x}^{\prime }}\right) - {g}_{n}\left( {x}^{\prime }\right) }\right| + \left| {{T}_{f}\left( {a, x}\right) - {g}_{n}\left( x\right) }\right| \] \[ < \frac{2}{n}\text{.} \] Letting \( {x}^{\prime } \) approach \( x \), we see that \[ \left| {{T}_{f}\left( {a,{x}^{ + }}\right) - {T}_{f}\left( {a, x}\right) - {g}_{n}\left( {x}^{ + }\right) + {g}_{n}\left( x\right) }\right| \leq \frac{2}{n}. \] Now noting that \[ {g}_{n}\left( {x}^{ + }\right) - {g}_{n}\left( x\right) = \left| {f\left( {x}^{ + }\right) - f\left( x\right) }\right| , \] we obtain \[ {T}_{f}\left( {a,{x}^{ + }}\right) - {T}_{f}\left( {a, x}\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}\left( {{g}_{n}\left( {x}^{ + }\right) - {g}_{n}\left( x\right) }\right) = \left| {f\left( {x}^{ + }\right) - f\left( x\right) }\right| . \] The rest of the proof is left as an exercise. The preceding proposition enables us to prove a result that is used later (in Theorem (6.1.18)) to establish the uniqueness of the representation of certain continuous linear functions. (1.5.19) Proposition. Let \( \alpha \) be a function of bounded variation on \( I = \) \( \left\lbrack {a, b}\right\rbrack \), and let \( D \) be the set consisting of \( a, b \), and all points of \( \left( {a, b}\right) \) at which \( \alpha \) is discontinuous. Then \( {\int }_{a}^{b}f\left( x\right) \mathrm{d}\alpha \left( x\right) = 0 \) for each continuous function \( f : I \rightarrow \mathbf{R} \) if and only if \( \alpha \left( x\right) = \alpha \left( a\right) \) for all \( x \in I \smallsetminus D \) . Proof. Note that \( D \) is countable, by Exercise (1.5.15:5). Suppose first that \( {\int }_{a}^{b}f\left( x\right) \mathrm{d}\alpha \left( x\right) = 0 \) for each continuous \( f \) on \( I \), and consider any point \( \xi \in I \smallsetminus D \) . There exist arbitrarily small