GRAPH MINORS AND HADWIGER’S CONJECTURE DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Eliade Mihai Micu, M.S. The Ohio State University 2005 Dissertation Committee: Approved by Professor Neil Robertson, Adviser Professor Thomas Dowling Adviser Professor Dijen Ray-Chaudhuri Graduate Program in Mathematics ABSTRACT One of the central open problems in Graph Theory is Hadwiger’s Con- jecture, which states that any graph with no Kk+1-minor is k-colorable. Re- stated, the conjecture asserts that the clique-minor number is always an upper bound for the chromatic number. In this paper we study various connections between these invariants. We start by providing the definitions and basic results needed later on, together with a new result about coloring ”almost all” the vertices of a graph. In the second chapter, we focus on graphs with stability number equal to two, proving that if such a graph does not contain an induced C4 or an induced C5, it satisfies Hadwiger’s Conjecture. The next chapter is dedicated to proving a result which is implied by the conjecture, i.e. an inequality linking the clique-minor numbers of a graph and its complement. We conclude the paper with a result about the embedding of any finite graph in Euclidean 3-space such that all its edges are straight line segments of integer length. ii In the memory of my mother iii ACKNOWLEDGMENTS Any writing is the product not only of its authors, but also of the envi- ronment where the authors work, of the encouragements and critics gathered from colleagues and teachers and conversations after seminars and confer- ences. While I cannot do justice to all of the above, I thank explicitly my adviser Dr. Neil Robertson for his encouragement, intellectual support and guidance throughout the years. He introduced me to the field of Graph Theory and suggested many deep and stimulating problems. I also want to thank Yung-Chen Lu for bringing me to The Ohio State University and helping me to get started in Graduate School. My perception of Combinatorics and Mathematics in general has devel- oped under the influence of many people. Especially, I am pleased to mention Thomas Dowling, Dijen Ray-Chaudhuri, Neil Falkner and John Hsia. Finally, my ultimate gratitude goes towards my wife, whose tremendous love, help and support made this work possible. iv VITA January 29, 1975 . Born - Constanta, Romania 1997 ...................... B.S. Mathematics, University of Bucharest 1999 ...................... M.S. Mathematics, University of Bucharest 2000 - present . Graduate Teaching Assistant, The Ohio State University FIELDS OF STUDY Major Field: Mathematics v TABLE OF CONTENTS ABSTRACT ................................. ii DEDICATION ................................ iii ACKNOWLEDGMENTS ........................ iv VITA ...................................... v LIST OF FIGURES ............................ viii 1 INTRODUCTION ........................... 1 1.1 BasicConcepts .......................... 1 1.2 GraphColorings.......................... 8 1.3 Graph Minors and Hadwiger’s Conjecture . 14 1.4 Large Clique Minors and Connections with Other Invariants . 17 1.5 UpperBoundsfortheChromaticNumber . 25 vi 2 HADWIGER’S CONJECTURE FOR GRAPHS WITH STA- BILITY NUMBER 2 ......................... 28 2.1 Preliminaries ........................... 28 2.2 Lower Bounds for the α =2Case ................ 31 2.3 Graphs with Low Connectivity . 34 2.4 ExcludingCertainSubgraphs . 36 2.5 GraphswithHighEdgeDensity . 43 3 GRAPH COMPLEMENTS ..................... 53 3.1 Preliminaries ........................... 53 3.2 Clique Minors in Graph Complements . 59 3.3 Complements of C4-freeGraphs ................. 61 3.4 Excluding Clique Minors in the Complement . 62 4 GRAPHS WITH EDGES OF INTEGER LENGTH .... 65 4.1 Preliminaries ........................... 66 4.2 TheMainResult ......................... 71 5 CONCLUSION AND FURTHER RESEARCH ....... 77 BIBLIOGRAPHY ............................. 80 vii LIST OF FIGURES 1.1 ConstructingaUniversalSubgraph. 22 2.1 The C4-freeCase.......................... 41 3.1 The values of f and F for n 5. ................ 57 ≤ 4.1 Embedding Kn into3-Space.. 75 viii CHAPTER 1 INTRODUCTION 1.1 Basic Concepts We begin this chapter by introducing the main concepts and by stating some of the classical Graph Theory results which will be used later. We denote by N the set of natural numbers, including zero, and by N∗ the set of all positive integers. For a real number x, the greatest integer less than or equal to x is denoted by x , and the least integer greater than or ⌊ ⌋ equal to x by x . ⌈ ⌉ For any set S and positive integer k, a family = S ,S , ,Sk of S { 1 2 ··· } k disjoint subsets of S is called a partition of S if S = i=1 Si. The sets S ,S , ,Sk are called the classes of the partition. WeS denote by k(S) 1 2 ··· P the set of all the subsets of S of cardinality k. 1 Definition 1.1.1 A graph is a triple G = (V,E,I), where V and E are disjoint sets and I is a mapping I : E (E) (E). →P1 ∪P2 The elements of V are called the vertices of the graph, the elements of E are called the edges of the graph. An edge e E is said to be a loop if ∈ I(e) (E); two distinct edges e,e′ E are called parallel if I(e)= I(e′). ∈P1 ∈ A graph is called finite if V is a finite set, it is called null if V and E are empty, and it is called simple if I is an injective mapping and its image is a subset of (V ). Unless otherwise stated, all the graphs considered in P2 this paper will be assumed to be non-null, finite and simple. For a simple graph G, we can view its edge set as a subset of (V ). In this case, we write P2 G =(V,E). A vertex v is said to be incident with an edge e if v I(e). For simple ∈ graphs we write e = uv if I(e) = u, v . The two vertices incident with { } an edge are called its endpoints. Two vertices u and v are adjacent or neighbors if uv E; two distinct edges are adjacent if they have a common ∈ endpoint. Let G =(V,E) and G′ =(V ′,E′) be two graphs. We say that G and G′ are isomorphic, and we write G G′, if there exists a bijection ϕ : V V ′ ≃ → such that xy E if and only if ϕ(x)ϕ(y) E′. Such a mapping ϕ is called ∈ ∈ an isomorphism. We set G G” := (V V ′,E E′) and G G” := (V V ′,E E′). If ∪ ∪ ∪ ∩ ∩ ∩ G G′ is the null graph, we say that G and G′ are disjoint. If two graphs ∩ H and K are disjoint, the graph G = H K is called the disjoint union of ∪ H and K. 2 If V ′ V and E′ E, we say that G′ is a subgraph of G and we write ⊆ ⊆ G′ G. If G′ contains all the edges uv E with u, v V ′, we say that G′ ⊆ ∈ ∈ is an induced subgraph of G. Definition 1.1.2 An independent (or stable) set is a set of pairwise non- adjacent vertices of the graph G. The size of the largest such set is called the independence (or stability) number of the graph and is denoted by α(G). Definition 1.1.3 A clique in a graph G is a set of pairwise adjacent ver- tices. If V (G) is a clique, the graph G is called complete. The size of the largest clique contained in a graph is called its clique number and is denoted by ω(G). For a vertex v V (G), its neighborhood in G, denoted by NG(v), is ∈ the set of all the vertices adjacent to v. If the reference is clear, we may drop the index referring to the underlying graph. More generally, for a set of vertices S V (G), the set of neighbors in V (G) S of vertices in S is called ⊆ \ the neighborhood of S and is denoted by NG(S). The degree dG(v) of a vertex v V (G) is the number of edges incident ∈ with v; for simple graphs, it is also the cardinality of NG(v). The minimum degree of G, denoted by δ(G), is defined to be δ(G) := min dG(v) : v V (G) . { ∈ } Similarly, the number ∆(G) := max dG(v) : v V (G) is called the maxi- { ∈ } mum degree of the graph G. 3 Definition 1.1.4 The average degree of a graph G is given by: 1 d(G) := d (v). V (G) G ∈ | | v XV (G) Remark 1.1.5 Clearly δ(G) d(G) ∆(G). ≤ ≤ Definition 1.1.6 A path is a graph P =(V,E) of the form: V = v ,v , ,vk and E = v v ,v v , ,vk− vk , { 0 1 ··· } { 0 1 1 2 ··· 1 } where the vi’s are all distinct. The path P defined above is denoted by P = v v vk. The vertices v 0 1 ··· 0 and vk are called the ends of the path P . The number of edges in a path is its length and a path of length k is denoted by Pk. If P = v v vk− is a path with k 3, then the graph C = P vk− v 0 1 ··· 1 ≥ ∪ 1 0 is called a cycle and we also write C = v v vk− . The length of the 0 1 ··· 1 cycle is the number of its edges; a cycle of length k is denoted by Ck. An edge not contained in a cycle, but joining two vertices of the cycle is called a chord. A cycle with no chords is called induced. Definition 1.1.7 A non-empty graph G is called connected if any two of its vertices can be joined by a path.