Large Structures in Dense Directed Graphs by Tássio Naia dos Santos A thesis submitted to the University of Birmingham for the degree of Doctor of Philosophy School of Mathematics College of Engineering and Physical Sciences University of Birmingham July 2018 University of Birmingham Research Archive e-theses repository This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder. If you have an apple and I have an apple and we exchange apples, then you and I will still each have one apple. But if you have an idea and I have an idea and we exchange these ideas, then each of us will have two ideas. George Bernard Shaw (attributed) Abstract We answer questions in extremal combinatorics, for directed graphs. Specifically, we investigate which large tree-like directed graphs are contained in all dense directed graphs of large order. More precisely, let T be an oriented tree of order n; among others, we establish the following results. (1) We obtain a sufficient condition which ensures every tournament of order n contains T , and show that almost every tree possesses this property. (2) We prove that for all positive C, ε and sufficiently large n, every tournament of order (1 + ε)n contains T if ∆(T ) ≤ (log n)C . (3) We prove that for all positive ∆, ε and sufficiently large n, every di- rected graph G of order n and minimum semidegree (1/2 + ε)n contains T if ∆(T ) ≤ ∆. (4) We obtain a sufficient condition which ensures that every directed graph G of order n with minimum semidegree at least (1/2 + ε)n contains T , and show that almost every tree possesses this property. (5) We extend our method in (4) to a class of tree-like spanning graphs which includes all orientations of Hamilton cycles and large subdivisions of any graph. Result (1) confirms a conjecture of Bender and Wormald and settles a con- jecture of Havet and Thomassé for almost every tree; (2) strengthens a result of Kühn, Mycroft and Osthus; (3) is a directed graph analogue of a classical result of Komlós, Sárközy and Szemerédi and is implied by (4); and (5) is of independent interest. Acknowledgements If this thesis ever s.aw the light of day, it is because I could stand in the shoulders of giants—Big Friendly Giants with whom I have had the honour to. share ideas, meals, dreams and worries, badminton games, laughs and so much more. I would like to thank you, whose supp. ort over the years has brought me so much joy and inspiration, and taught me more than I can hope to acknowledge. First and foremost, I would like to thank my advisor, Richard, for his knowledge, understanding, kindness and support throughout the PhD, and for inspiring me to do more and better mathematics. I would also like to thank Andrew and Julia for the careful reading of this thesis and for their many insightful suggestions. My life has been enriched at and beyond work by many wonderful people I h. ave met at Birmingham. My sincere thanks go to the Mathematics department, especially the comb.inatorics group—present and past—for be.ing such a vibrant and warm community, and for many stimulating conversations. My PhD would not have been nearly as fulfilling if it wasn’t for some special friends. In a pseudo-random order (promise!) Frederik, Stefan, Fábio, Robert, Andrew, Stefano, Nicolás, Ryan and Ben. This se.ction would be incomplete without including André and Johanna (and family!) for your warm welcome and for being home away from home. Rent can be expensive if you have a family but no salary. I am tru. ly grateful for the financial support provided by CNPq (Proc. 201114/2004-3), however late it came. I have been blessed with a wonderful family, who supports me well beyond reason. Special thanks go to my parent.s, Adonel and Célia, for giving me wings and roots. Vocês me deram carinho, educação, e sempre acreditaram em mim. Elisabeth and Kale, thank you for be.ing there always, especially in times of need. Emmanuelle and Tamira, for your rooting and deep ca.ring. I love you all. And of course, Sophie and Ka’imi. Words can’t do you justice, but all the people above know how m. uch you mean to me. You are always there for me—in so. many ways, especially when it is hard. Je vous aime à la folie, de montão. I am better, every day, because of you. Contents Contentsi List of Theorems iii List of Figures vii 1 Introduction1 1.1 Unavoidable trees.......................... 1 1.2 Spanning trees via high semidegree................ 6 1.3 Trees via chromatic number.................... 11 2 Preliminary concepts and results 17 2.1 Trees................................. 19 2.2 Regularity.............................. 23 2.3 Tournaments ............................ 25 2.4 Useful estimates and bounds.................... 26 2.5 Homomorphisms, allocation, embedding ............. 28 2.5.1 Allocation.......................... 29 2.5.2 Embedding ......................... 30 3 Spanning Trees of Tournaments 31 3.1 Proof outline for Theorem 1.4................... 31 3.2 Preliminaries ............................ 34 3.3 An approximate result (Theorem 1.9)............... 35 3.3.1 Allocation around a cycle of cluster tournaments . 35 3.3.2 Embedding around a cycle of cluster tournaments . 37 3.3.3 Proof of Theorem 1.9.................... 41 3.4 Almost-directed pairs........................ 50 3.5 Cycles of cluster tournaments................... 55 3.5.1 Embedding the first subtree................ 55 3.5.2 Embedding the second subtree............... 58 3.5.3 Joining the pieces...................... 66 3.6 Unavoidable trees (Theorems 1.4 and 1.6) ............ 69 i ii Contents 3.6.1 A class of unavoidable trees (Theorem 1.4) . 69 3.6.2 Most oriented trees are nice (Theorem 1.6) . 71 4 Spanning Structures via Semidegree 75 4.1 Proof outline for Theorem 1.13 .................. 75 4.2 Preliminaries ............................ 77 4.2.1 Bare paths.......................... 77 4.2.2 Regularity.......................... 78 4.2.3 Matchings.......................... 81 4.2.4 Diamond-paths....................... 82 4.3 An approximate result....................... 85 4.3.1 Existence of regular expander subdigraph......... 86 4.3.2 Random walks ....................... 88 4.3.3 Allocation algorithm.................... 91 4.3.4 Embedding ......................... 94 4.3.5 An approximate result (Theorem 4.17).......... 98 4.4 Trees with many bare paths.................... 99 4.4.1 Allocation.......................... 99 4.4.2 Proof of Lemma 4.20.................... 105 4.5 Trees with many leaves....................... 110 4.5.1 Allocation.......................... 111 4.5.2 Proof of Lemma 4.23.................... 114 4.6 Proof of Theorems 1.12, 1.13 and 1.15 . 118 4.6.1 Tree-like spanning subdigraphs . 118 5 Subdigraphs via chromatic number 125 5.1 Typical behaviour.......................... 125 5.2 Non-probabilistic results...................... 129 5.2.1 Burr’s conjecture holds for stars . 131 6 Further directions 133 6.1 Trees in tournaments........................ 133 6.2 Trees in digraphs via semidegree.................. 136 6.3 Trees via chromatic number.................... 137 Index of Definitions 139 Bibliography 141 List of Theorems Every algorithm, claim, conjecture, corollary, fact, definition, lemma, problem, question, and theorem in the thesis, with a brief description and references when ob- tained from other sources. Main results are starred. Definitions are indexed on page 140. I 1 Introduction t 1.1 Almost every tree is almost unavoidable[8].......................................... 3 ? t 1.2 Almost every tree is unavoidable.................................................... 3 d 1.3 α-nice trees......................................................................... 4 ? t 1.4 Nice trees with polylogarithmic maximum degree are unavoidable.................... 4 t 1.5 Almost every tree has sublogarithmic maximum degree[65].......................... 5 t 1.6 Almost every tree is 1/250-nice...................................................... 5 conj 1.7 All tournaments of order 2n − 2 contain every tree of order n [83].................... 5 conj 1.8 All tournaments of order n + ` − 1 contain every tree with order n and ` leaves[39].. 5 ? t 1.9 Approximate embedding of T with polylog ∆(T ) in all tournaments................. 6 conj 1.10 Caccetta–Haggkvist’s conjecture[17]................................................. 7 t 1.11 Komlós, Sárközy and Szemerédi’s theorem[49]....................................... 8 ? t 1.12 Large semidegree guarantees every bounded-degree spanning tree.................... 9 ? t 1.13 Sufficient condition for T to be spanning subtree of high-semidegree G .............. 10 ? t 1.14 If G has high semidegree then G contains almost every spanning tree............... 10 ? t 1.15 Tree-like spanning subdigraphs of bounded degree.................................. 11 t 1.16 There exist graphs with large girth and chromatic number[5]....................... 11 conj 1.17 If T has order n then every orientation of a (2n − 2)-chromatic graph contains T ....12 t 1.18 Every orientation of a graph contains a directed path of order χ(G) [32, 38, 69, 82]. 12 |T | t 1.19 If T is an oriented tree and D is a digraph with χ(D) ≥ 2 + 1 then T ⊆ D [1].... 13 t 1.20 There exists a tournament containing no acyclic subgraph H with ‘large’ χ(H) ..... 13 t 1.21 Trees in random orientations of graphs via minimum degree........................ 13 ? t 1.22 Every orientation of G contains every tree of order χ(G)/(1 + log2 |G|).............. 14 ? conj 1.23 If T is an oriented tree, then t(T ) = q(T ) ..........................................