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Extremal Questions in Graph Theory Extremal questions in graph theory vorgelegt von Maya Jakobine Stein Habilitationschrift des Fachbereichs Mathematik der Universit¨atHamburg Hamburg 2009 Contents 1 Introduction 1 2 The Loebl–Koml´os–S´osconjecture 5 2.1 History of the conjecture . 5 2.2 Special cases of the Loebl–Koml´os–S´osconjecture . 6 2.3 The regularity approach . 6 2.4 Discussion of the bounds . 7 3 An approximate version of the LKS conjecture 9 3.1 An approximate version and an extension . 9 3.2 Preliminaries . 10 3.2.1 Regularity . 10 3.2.2 The matching . 13 3.3 Proof of Theorem 2.3.3 . 15 3.3.1 Overview . 15 3.3.2 Preparations . 18 3.3.3 Partitioning the tree . 20 3.3.4 The switching . 23 3.3.5 Partitioning the matching . 25 3.3.6 Embedding lemmas for trees . 28 3.3.7 The embedding in Case 1 . 33 3.3.8 The embedding in Case 2 . 34 3.4 Proof of Theorem 3.1.1 . 36 4 Solution of the LKS conjecture for special classes of trees 39 4.1 Trees of small diameter and caterpillars . 39 4.2 Proof of Theorem 4.1.1 . 40 4.3 Proof of Theorem 4.1.2 . 45 5 An application of the LKS conjecture in Ramsey Theory 49 5.1 Ramsey numbers . 49 5.2 Ramsey numbers of trees . 50 5.3 Proof of Proposition 5.2.2 . 51 6 t-perfect graphs 53 6.1 An introduction to t-perfect graphs . 53 6.2 The polytopes SSP and TSTAB . 55 6.3 t-perfect line graphs . 56 6.4 Squares of cycles . 59 6.5 The main lemma . 60 6.6 Colouring claw-free t-perfect graphs . 63 6.7 Characterising claw-free t-perfect graphs . 66 2 2 6.8 C7 and C10 are minimally t-imperfect . 70 7 Strongly t-perfect graphs 75 7.1 Strong t-perfection . 75 7.2 Strong t-perfection and t-minors . 76 7.3 Strongly t-perfect and claw-free . 78 7.4 Minimally strongly t-imperfect . 85 8 Infinite extremal graph theory 89 8.1 An introduction to infinite extremal graph theory . 89 8.2 Terminology for infinite graphs . 91 8.3 Grid minors . 92 8.4 Connectivity of vertex-transitive graphs . 95 9 Highly connected subgraphs of infinite graphs 97 9.1 The results . 97 9.2 End degrees and more terminology . 99 9.3 Forcing highly edge-connected subgraphs . 99 9.4 High edge-degree but no highly connected subgraphs . 104 9.5 Forcing highly connected subgraphs . 107 9.6 Linear degree bounds are not enough . 112 10 Large complete minors in infinite graphs 117 10.1 An outline of this chapter . 117 10.2 Large complete minors in rayless graphs . 118 10.3 Two counterexamples . 118 10.4 Large relative degree forces large complete minors . 120 10.5 Using large girth . 123 11 Minimal k-(edge)-connectivity in infinite graphs 125 11.1 Four notions of minimality . 125 11.2 The situation in finite graphs . 126 11.3 The situation in infinite graphs . 128 11.4 Vertex-minimally k-connected graphs . 132 11.5 Edge-minimally k-edge-connected graphs . 135 11.6 Vertex-minimally k-edge-connected graphs . 137 12 Duality of ends 141 12.1 Duality of graphs . 141 12.2 The cycle space of an infinite graph . 143 12.3 Duality for infinite graphs . 145 12.4 Discussion of our results . 145 12.5 induces a homeomorphism on the ends . 148 ∗ 12.6 Tutte-connectivity . 150 12.7 The dual preserves the end degrees . 155 vi Chapter 1 Introduction Extremal graph theory is a branch of graph theory that seeks to explore the properties of graphs that are in some way extreme. The classical extremal graph theoretic theorem and a good example is Tur´an'stheorem. This theo- rem reveals not only the edge-density but also the structure of those graphs that are `extremal' without a complete subgraph of some fixed size, where extremal means that upon the addition of any edge the forbidden subgraph will appear. This is the type of question we study in extremal graph the- ory, for finite as well as for infinite graphs. In general one asks whether some invariant { which instead of the edge-density might be the minimum degree, or the chromatic number, etc { has an influence on the appearance of substructures, or on another graph invariant. A typical example of modern extremal graph theory is the Loebl–Koml´os{ S´osconjecture (LKS-conjecture for short) from 1992. This conjecture is about whether the existence of subtrees in a graph can be forced by as- suming a large median degree. More precisely, the LKS-conjecture states that every graph G that has at least G =2 vertices of degree at least some j j k N, contains as subgraphs all trees with k edges. We shall discuss the LKS-conjecture2 and steps towards a solution, as well as an application of the conjecture in Ramsey theory, in Chapters 2, 3, 4 and 5. Our main contribution to this field is an approximate version of the LKS-conjecture, which will be presented in Chapter 3. The proof of this version builds on work of Ajtai, Koml´osand Szemer´edi[1] and features an application of Szemer´edi'sregularity lemma. This chapter is based on work from [74]. In Chapter 4, which is based on work from [75], we shall solve the LKS- conjecture for special classes of trees. In Chapter 5 we discuss the impact of the conjecture in Ramsey theory. 1 2 Introduction A theory belonging to extremal graph theory in a broader sense will be the topic of Chapters 6 and 7. The much studied perfect graphs (those for which the chromatic number of every induced subgraph H equals the clique number of H) were introduced by Berge in the early 1960's. One can characterise perfect graphs in terms of their stable set polytope (SSP for short), and in this context Chv´atal[25] proposed the concept of t-perfect graphs. These are defined via properties of their SSP, in fact, by a slight modification of the properties which the SSP of a perfect graph must have, involving a second polytope, namely TSTAB. We will be concerned with colourings and characterisations of t-perfect graphs by forbidden t-minors. So, if the relation of the SSP and the TSTAB of a graph (both to be formally defined in Chapter 6) is viewed as one of the graph's invariants, then what we are studying is this invariant's impact on the chromatic number, and if we should succeed in a characterisation, we would indeed force substructure via an invariant. We discuss t-perfect and the closely related strongly t-perfect graphs in Chapters 6 and 7, which are based on [13, 15]. In general, extremal graph theory has been a very active area of graph theory during the last decades. However, until recently, an extremal branch of infinite graph theory did practically not exist. The reason for this is that in general, the behaviour of infinite graphs is not as well understood as the behaviour of finite graphs. Often it is not clear how certain invariants translate `correctly' to infinite graphs. For example, how does a condition on the edge-density translate to an infinite graph? How should one define the average degree of an infinite graph? Even for parameters that appear to have an obvious counterpart in in- finite graph theory, it may happen that they lose the power they have in finite graphs. One example is the minimum degree. In finite graphs, a high minimum degree can imply the existence of large complete subgraphs (this is a corollary of Tur´an'stheorem mentioned above). But in infinite graphs, the assumption of a high minimum degree loses its strength, as any minimum degree condition can be met by an infinite tree which is not `dense' enough to contain an interesting substructure. A solution to this dilemma are the end degrees, to be defined in Chapter 8 (and a variant in Chapter 10), which, if large enough, provide a certain denseness `at infinity' and thus make it possible to force substructure in infinite graphs. The most interesting of these substructures are doubtlessly large complete minors. With weaker assumptions, we can still force highly connected subgraphs and grid minors. These and related topics will be the subject of Chapters 8, 9 and 10, which are based on work from [83, 86]. 3 We shall encounter another application of the end degrees in infinite ex- tremal graph theory in Chapter 11. The topic of this chapter, which is based on work from [82], are minimally k-connected graphs. For finite graphs, mainly two types of minimality have been investigated: minimality with re- spect to edge-deletion, which we shall call edge-minimality, and with respect to vertex-deletion, which we shall call vertex-minimality (In the literature, one often encounters the terms minimality for the former and criticality for the latter type). In finite such graphs bounds on the minimum degree and on the number of vertices which attain it have been much studied. We give an overview of the results known for finite graphs and show that basically all of the results carry over to infinite graphs if we consider ends of small degree as well as vertices. Chapter 12 is on duality of infinite graphs. This chapter is based on work from [14]. The main interest will be the end space of a graph in relation to the ends space of its dual.
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