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UNIVERSITY of CALGARY Graph Colouring and Forbidden UNIVERSITY OF CALGARY Graph Colouring and Forbidden Subgraphs by Jia Shen A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS AND STATISTICS CALGARY, ALBERTA APRIL, 2008 °c Jia Shen 2008 ISBN: 978-0-494-38239-4 Abstract Concepts and questions of graph colouring arise naturally from practical problems and have found applications in many areas, including Information Theory and most notably Theoretical Computer Science. This thesis mainly concerns graph colouring problems. Given a graph F , a graph H is said to be F -free if H does not contain F as an induced subgraph. We will be concerned with what we call the F -avoiding colouring problem: for a given graph F , colour the vertices of a graph G such that each maximal F -free subgraph of G receives more than one colour. This colouring is motivated by questions raised by Erd}os-Gallai-Tuza, and Duffus-Kierstead-Trotter. The following question will be resolved in full: Given a graph F , does there exist a constant c, depending only on F , such that every graph G has a vertex-colouring with at most c colours such that each nontrivial maximal F -free subgraph of G receives more than one colour? Using a similar technique, the related transversal problem is also completely resolved. We also investigate the precise number of colours needed for various classes of graphs F and G. The corresponding problems in the setting of partially ordered sets are studied as well. iii Acknowledgements First I thank Dr. Bill Sands for accumulating discussions towards the results in this thesis. Dr. Sands read through the draft of this thesis many times and corrected many of the inaccuracies. I also would like to thank Dr. Sands for lending me some precious books in his collection, which are very helpful. Thanks also go to the faculty and the staff of the Department of Mathematics and Statistics at the University of Calgary for all the help I got during the past years. Finally, I would like to thank my dear parents and my beloved wife, Jingjing Zhou. It is your encouragement, patience, support and love that made my studies possible. iv Table of Contents Abstract iii Acknowledgements iv Table of Contents v 1 Introduction 1 1.1 Subject of Research . 1 1.2 Motivation and Historical Notes . 2 1.3 Relations and Differences . 5 1.4 Structure of the Thesis . 7 2 Definitions, Notation and Basic Facts 8 2.1 The Basic Notation . 8 2.2 Graphs . 8 2.3 Graph Colouring . 12 2.4 Hypergraphs . 13 2.5 Basic Properties of F -avoiding Colouring . 15 3 Clique Colouring 21 3.1 Constant Bounds . 21 3.2 General Bounds . 24 3.3 Complexity . 25 3.4 Tree-decompositions and Chordal Graphs . 26 v vi 4 A Complete Characterization 31 4.1 Bounded Avoiding Chromatic Numbers . 31 4.2 Unbounded Avoiding Chromatic Numbers . 33 4.3 General Bounds . 38 4.4 Transversals . 44 5 Graphs F without Isolated Vertices 51 5.1 General Results . 51 5.2 Various Classes of Graphs . 60 6 Small F with Isolated Vertices 70 6.1 Characterization of Maximal Subgraphs . 70 6.2 Union of Cliques . 71 6.3 Sparse Graphs . 74 6.4 Induced Long Path . 77 6.5 Some Other Classes of Graphs . 80 7 Partially Ordered Sets 93 7.1 Preliminaries for Partially Ordered Sets . 93 7.2 Colouring of Posets . 98 7.3 Bounds . 100 7.4 Decomposition of Partially Ordered Sets . 102 7.5 Relations between Various Bounds . 109 7.6 An Approach towards Two Colourable Posets . 117 8 Remarks and Open Problems 126 vii 8.1 Open Problems . 126 8.2 Classes of Graphs F . 128 Bibliography 130 List of Figures 2.1 A disjoint union of Kk's. 18 3.1 a tree-decomposition (T; ) = (T; (V ) 2 ) . 27 V t t T 3.2 a tree-decomposition of G ........................ 29 4.1 F1 F2 and an edge . 35 _ 4.2 3-cycle v1e1ve2v2ev1 ............................ 37 4.3 colouring scheme . 39 5.1 diam(G) 3................................ 52 ≥ 5.2 the graph K3 + e ............................. 54 5.3 K2 and a vertex . 55 5.4 A list of graphs: K ;P ;K ;P ;C ;K + e; K e . 56 2 3 3 4 4 3 4 − 5.5 N[v] and a vertex x ............................ 57 5.6 F is a star and G is not a star . 62 5.7 The colouring of wheel Wn ........................ 66 6.1 acP3 (G) for union of cliques . 71 6.2 ac (G) where g(G) 5 ......................... 75 P3 ≥ 6.3 acP3 (P5).................................. 80 6.4 acP3 (Pk).................................. 81 6.5 acP3 (Cn).................................. 82 6.6 acP3 (G) for bigraphs G with δ(G)=1.................. 87 7.1 A partition of a linearly indecomposable poset . 103 viii ix 7.2 the 17-element poset Q . 112 7.3 A poset P with ack(P ) = 3 . 113 7.4 Example 7.1 . 116 7.5 the relation m n . 119 ≺Q 7.6 a1 Q a2 Q Q ak Q a1 . 120 2k+1≺ ≺ · · · ≺ ≺ 7.7 ( 2)-free . 121 X1 k 7.8 ( n)-free . 123 X1 k 7.9 ( (2 1))-free . 124 ⊕ X1 k 7.10 ( (1 2))-free . 125 ⊕ X1 Chapter 1 Introduction In this thesis, all graphs are finite, undirected and simple (i.e., contain no multiedges and loops) unless otherwise specified. 1.1 Subject of Research When considering relational structures such as graphs and partially ordered sets, subsets which do not contain a prescribed, induced \forbidden" substructure are often of interest. For example, if the forbidden subgraph is K2 (i.e. two independent vertices), then subgraphs that do not contain K2 are all complete subgraphs. Thus a maximal K2-free subgraph is a maximal complete subgraph, or a maximal clique. In general, for a given graph F , a graph is said to be F -free if it does not contain F as an induced subgraph. The main subject of this thesis is to investigate, for a given \forbidden" graph F , maximal F -free subgraphs of a graph G. Our main interest will be what we call the F -avoiding colouring problem1: for a given graph F , colour the vertices of a graph G such that each maximal F -free subgraph of G receives more than one colour. We will be concerned with the following question: Question 1: Given a graph F , does there exist a constant c, depending only on F , 1The term \F -free colouring" has been defined by other researchers for a different, yet related, colouring problem. We will review this concept in Section 1.4. 1 2 such that every graph G has a vertex-colouring with at most c colours so that each nontrivial (i.e. with more than one vertex) maximal F -free subgraph of G receives more than one colour? A related question is the F -free subgraph transversal: a subset T V (G) of ⊆ vertices of a graph G is said to be an F -free subgraph transversal if T meets every maximal F -free subgraph of G. For a given \forbidden" graph F , the F -free subgraph transversal number of a graph G, denoted by τF (G), is the minimum cardinality of an F -free subgraph transversal of G. We will also be considering the following question: Question 2: For which graphs F is the F -free subgraph transversal number τF (G) bounded by a constant fraction of the order of G, namely, τ (G) c G F ≤ j j for some constant c < 1 independent of G? 1.2 Motivation and Historical Notes The subject of this thesis is motivated by a problem raised by Erd}os,Gallai and Tuza. In [30] Erd}os, Gallai and Tuza proposed to investigate the so-called clique- transversal problem, namely, the problem of estimating the minimal cardinality of a subset of V (G) which meets every maximal clique of G. In particular, according to Erd}os(cf. [2, 24]), Gallai asked whether a triangulated graph (or chordal graph) on n vertices has a subset of size n=2 which intersects every nontrivial maximal clique. In fact, the following stronger result is true: The vertices of a chordal graph can be 3 2-coloured so that all nontrivial maximal cliques receive two colours. We will give a proof of it in Chapter 3. Aigner and Andreae [2] noted that, as can be easily seen, a comparability graph G has a subset of size at most G =2 which intersects every nontrivial maximal clique. j j They also raised the question: is this true for a cocomparability graph? Lonc and Rival [38] conjectured something stronger: a cocomparability graph is 2-colourable so that each nontrivial maximal clique receives both colours. If this statement were true, then the smaller of the two colour classes would have size at most n=2 and would intersect every maximal clique. Duffus, Sands, Sauer, and Woodrow gave a counterexample in [25] to both ques- tions. However, Duffus, Kierstead and Trotter [24] proved that the next best thing to the Lonc-Rival conjecture is true, i.e., three colours instead of two are enough. The vertices of a cocomparability graph can be 3-coloured so that all nontrivial maximal cliques receive at least two colours. In the language of partially ordered sets, what Duffus et al. have proven is that the elements of any partially ordered set can be 3-coloured so that all nontrivial maximal antichains receive more than one colour.
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