Chromatic and Structural Properties of Sparse Graph Classes Daniel A. Quiroz Supervisor: Jan van den Heuvel A thesis presented for the degree of Doctor of Philosophy Department of Mathematics London School of Economics and Political Science 1 Declaration I certify that the thesis I have presented for examination for the MPhil/PhD degree of the London School of Economics and Political Science is solely my own work, with the following exceptions: • Chapter 2 is joint work with Jan van den Heuvel, Patrice Ossona de Mendez, Roman Rabinovich, and Sebastian Siebertz. It is based on published work [39]. • Chapter 3 is joint work with Jan van den Heuvel and Hal Kierstead. It is based on work accepted for publication [37]. • Chapter 5 is joint work with Jan van den Heuvel, Stephan Kreutzer, MichalPilipczuk, Roman Rabinovich, and Sebastian Siebertz. It is based on published work [38]. The copyright of this thesis rests with the author. Quotation from this thesis is permitted, provided that full acknowledgement is made. This thesis may not be reproduced without my prior written consent. I warrant that this authorisation does not, to the best of my belief, infringe the rights of any third party. 2 Abstract A graph is a mathematical structure consisting of a set of objects, which we call vertices, and links between pairs of objects, which we call edges. Graphs are used to model many problems arising in areas such as physics, sociology, and computer science. It is partially because of the simplicity of the definition of a graph that the concept can be so widely used. Nevertheless, when applied to a particular task, it is not always necessary to study graphs in all their generality, and it can be convenient to studying them from a restricted point of view. Restriction can come from requiring graphs to be embeddable in a particular surface, to admit certain types of decompositions, or by forbidding some substructure. A collection of graphs satisfying a fixed restriction forms a class of graphs. Many important classes of graphs satisfy that graphs belonging to it cannot have many edges in comparison with the number of vertices. Such is the case of classes with an upper bound on the maximum degree, and of classes excluding a fixed minor. Recently, the notion of classes with bounded expansion was introduced by Neˇsetˇriland Ossona de Mendez [62], as a generalisation of many important types of sparse classes. In this thesis we study chromatic and structural properties of classes with bounded expansion. We say a graph is k-degenerate if each of its subgraphs has a vertex of degree at most k. The degeneracy is thus a measure of the density of a graph. This notion has been generalised with the introduction, by Kierstead and Yang [47], of the generalised colouring numbers. These parameters have found applications in many areas of Graph Theory, including a char- acterisation of classes with bounded expansion. One of the main results of this thesis is a series of upper bounds on the generalised colouring numbers, for different sparse classes of graphs, such as classes excluding a fixed complete minor, classes with bounded genus and classes with bounded tree-width. We also study the following problem: for a fixed positive integer p, how many colours do we need to colour a given graph in such a way that vertices at distance exactly p get different colours? When considering classes with bounded expansion, we improve dramatically on the previously known upper bounds for the number of colours needed. Finally, we introduce a notion of addition of graph classes, and show various cases in which sparse classes can be summed so as to obtain another sparse class. 3 Acknowledgement \I would maintain that thanks are the highest form of thought, and that gratitude is happiness doubled by wonder." { Gilbert K. Chesterton It will soon be four years, since I started my \PhD studies". During this chapter of my life, I have been accompanied and sustained by many persons and, for this, I owe them immense gratitude. The fact that this chapter is summarised now, in the form of the present thesis, produces in me an indescribable mixture of feelings. I would like to thank my wife Maria, for never restraining her immense heart. Thank you for all your hard work, your patience and for accompanying me with so much care in everything I do. Although reading it may have a soporific effect on you, this thesis is as much yours as it is mine. Thanks also to my son Lucas { you and your mum are an oasis of sweetness and beauty. I would like to thank my parents for their love and sacrifice, and for introducing me to the beauty of mathematics and of life in general. To my brother Andr´es{ your are always with me despite the distance { and to my sister Isabel { it has been an immense blessing to have you near by during these years! A special thanks to my supervisor, Jan van den Heuvel. It has been an immense pleasure to get to know and work with such an amazing mathematician and great person. Thank you for your patience, your kindness, your sincere enthusiasm, for introducing me to the problems and theory which comprise this thesis, and for working with me on these. Thank you also for proofreading this thesis so many times. Thanks to all my other co-authors: Hal Kierstead, Stephan Kreutzer, Patrice Ossona de Mendez, Micha l Pilipczuk, Roman Rabinovich, and Sebastian Siebertz. Special thanks to Hal, Roman and Sebastian for stimulating discussions and for pointing to the right tools at the right time. Thanks to everyone in the Department of Mathematics. To all the academic staff, specially those in the Discrete Mathematics group. To my fellow PhD students, specially Ahmad, Barnaby, Bernhardt, Jos´eand Nicola. To the administrative staff, specially Becca and Kate. Thanks to the Department and the Financial Support Office for always stepping in when the going got tough. 4 Thanks to my examiners for giving my thesis such a careful reading, and for their com- ments and suggestions. Thanks to Mark, Meera, Becca, and all those who have welcomed us with their friendship in this country. Thanks also to all my friends and family, those who remain in Venezuela and those who have fled. Thanks to Danny, Ascenci´on,and all my grandparents for their constant prayers. Thanks to Saint Jude Thaddeus, patron of difficult and desperate causes, and to Saint Thomas Aquinas, patron of students, for their intercession. Thanks to the Most Blessed Virgin Mary, for her most sweet care and her constant inter- cession. Praise and thanks to the three persons of the Most Holy Trinity, for sustaining our exis- tence, and humanity through the Sacraments. Contents 1 Introduction 7 1.1 Subgraphs, minors and classes with bounded expansion . 9 1.2 Chromatic number, degeneracy and the generalised colouring numbers . 16 1.3 Graph powers and exact distance graphs . 20 1.4 Summary . 24 2 Generalised colouring numbers of graphs that exclude a fixed minor 27 2.1 Introduction . 27 2.2 Tree-width and generalised colouring numbers . 32 2.3 Flat decompositions . 35 2.4 The weak k-colouring numbers of graphs excluding a fixed complete minor . 39 2.5 The strong k-colouring numbers of graphs excluding a fixed minor . 41 2.6 The generalised colouring numbers of planar graphs . 46 2.6.1 The weak k-colouring number of planar graphs . 46 2.6.2 The strong k-colouring number of planar graphs . 48 3 Chromatic Numbers of Exact Distance Graphs 54 3.1 Introduction . 54 3.2 Exact distance graphs and generalised colouring numbers . 60 3.2.1 Proof of Theorem 3.1.7 . 60 3.2.2 Proof of Theorem 3.1.8 . 61 3.3 Explicit upper bounds on the chromatic number of exact distance graphs . 65 3.3.1 Graphs with bounded tree-width . 65 3.3.2 Graphs with excluded complete minors . 66 5 Contents 6 3.4 Lower bounds on the chromatic number of exact distance-3 graphs . 68 3.5 Lower bounds on the chromatic number of weak distance-p graphs . 70 4 Colouring exact distance graphs of chordal graphs 74 4.1 Introduction . 74 4.2 Level partitions of chordal graphs . 78 4.3 Adjacent-cliques graphs . 80 4.4 Exact distance graphs of chordal graphs . 81 4.5 A remark on graphs with bounded genus . 83 5 When are sparse graph classes closed under addition? 86 5.1 Introduction . 86 5.1.1 Adding edges, graphs, and classes . 86 5.1.2 k-admissibility and uniform orderings . 90 5.2 Classes with bounded tree-width plus the class of paths . 92 5.3 Page number and adding minor closed classes . 95 5.4 Adding spanning trees of small degree while preserving k-admissibility . 96 5.5 An application of Theorem 5.1.11 . 103 5.5.1 First-order logic . 103 5.5.2 Model-checking for successor-invariant first-order logic . 104 6 Open problems and further directions 108 6.1 Chromatic number of exact distance graphs . 108 6.1.1 Mind the gap . 108 6.1.2 Obtaining upper bounds for more graph classes . 109 6.1.3 Considering multiple odd distances . 111 6.2 Generalised colouring numbers . 112 6.3 Adding graph classes . 113 Bibliography 115 Chapter 1 Introduction In the same year as the Great Fire of London, a young Leibniz presented, as part of his doctoral thesis, his Dissertatio de arte combinatoria [58]. While the present thesis shares with that of Leibniz that it deals with problems which fall within the broad area of Combinatorics, we cannot claim, as Leibniz did in the front page of his dissertatio, that we \prove the existence of God with complete mathematical certainty".