Graph Minors and Hadwiger's Conjecture

Home , Four color theorem, Hadwiger number

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

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 vertices. 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.

If G is a graph, a maximal connected subgraph of G is called a component of G. A component is said to be odd if it contains an odd number of vertices, and it is said to be even otherwise.

If A and B are non-empty subsets of vertices of a graph G, we say that a set of vertices X separates A and B in G if any path linking a vertex in A

4 to a vertex in B contains a vertex from X. X is called a separating set for

A and B or a vertex cutset in G. A vertex separating two other vertices lying in the same connected component of G is called a cutvertex.

A graph G is called k-connected for some k N if V (G) > k and G X ∈ | | \ is connected for any set of vertices X with X < k. This implies that no | | two vertices of the graph are separated by fewer than k vertices. The largest integer k such that G is k-connected is the connectivity k(G) of G.

Deﬁnition 1.1.8 Let r 2 be an integer. A graph G is called r-partite if ≥ its vertex set admits a partition into r classes such that every edge has its endpoints in diﬀerent classes. If r = 2, such graphs are called bipartite.

In other words, it suﬃces to require that all the classes are independent sets. If any two vertices belonging to diﬀerent classes are adjacent in an r-partite graph, the graph is called complete. It is easy to prove that a graph is bipartite if and only if it contains no odd cycle.

Deﬁnition 1.1.9 A matching in a graph G is a set M of edges such that no two are adjacent. We say that M is a matching of S V (G) if every ⊆ vertex of S is incident with an edge of M.

5 Deﬁnition 1.1.10 A vertex cover C of G is a set of vertices such that any edge is incident with a vertex in C.

The maximal cardinality of a matching in a bipartite graph is given by the following:

Theorem 1.1.11 (K¨onig) In any bipartite graph G, the maximum cardinality of a matching is equal to the minimum cardinality of a vertex cover.

A matching M in a graph G is called perfect if any vertex of G is incident with an edge of M. For a graph G, let p(G) be the number of connected components of G and p1(G) be the number of odd components of G. The existence of a perfect matching in general graphs is described by the following:

Theorem 1.1.12 (Tutte) A graph G has a perfect matching if and only if p (G S) S for all S V (G). 1 \ ≤ | | ⊆

A non-empty graph G is called critical if G v has a perfect matching \ for any vertex v V (G). A vertex set S V (G) is called matchable to ∈ ⊆ G S if the bipartite graph arising from G by contracting all the connected \ components of G S to single vertices and deleting all the edges inside S, \ contains a matching of S. The deletion and contraction operations are deﬁned in Section 1.3.

6 The following theorem, whose proof can be found in Section 2.2 of [3], describes a matching structure on an arbitrary graph:

Theorem 1.1.13 Any graph G contains a set S of vertices with the following properties:

1. S is matchable to G S; \

2. every component of G S is critical. \

For any such set S, the graph G has a perfect matching if and only if S = p(G S). | | \

We conclude this section with a short description of Tur´an’s Theorem.

Deﬁnition 1.1.14 Let n be a positive integer and H be a graph. A graph

G on n vertices and not containing H as a subgraph is called extremal if it has the largest possible number of edges.

We denote by ex(n, H) the number of edges of an extremal graph G for

H and n. If H = Kr for some r 2, all the complete (r 1)-partite graphs ≥ − are edge-maximal without containing Kr. The unique (up to isomorphism) complete (r 1)-partite graphs on n r 1 vertices whose partition classes − ≥ − diﬀer in size by at most one are called Tur´an graphs and are denoted by

Tr−1(n). Also we denote by tr−1(n) the number of edges of Tr−1(n).

7 Theorem 1.1.15 (Tur´an) For all integers r, n with r 2, every graph G ≥ not containing Kr, with n vertices and ex(n, Kr) edges is a Tr−1(n).

Proof. See, for instance, Section 7.1 in [3].

The following upper bound will be useful later:

Proposition 1.1.16 For any integers r, n with r 2 we have: ≥

1 2 r 2 tr− (n) n − . 1 ≤ 2 r 1 −

Proof. Let A ,A , ,Ar− be the classes of an (r 1)-partition of Tr− (n). 1 2 ··· 1 − 1 We have:

1.2 Graph Colorings

Let G be a loopless graph, possibly with multiple edges. Let V (G) be the vertex set of G and E(G) be the edge set of G. For any positive integer k, let [k] denote the set 1, 2, , k . { ··· }

Deﬁnition 1.2.1 A vertex k-coloring of G is a mapping c : V (G) [k] → with the property that c(u) = c(v) if the vertices u and v are adjacent in G. 6 8 A vertex k-coloring is also referred to as a k-coloring or as a proper coloring of the vertices of a graph G. If G admits a k-coloring, we say that G is k-colorable. The set of all the vertices receiving the same color in such a coloring is called a color class. It is easy to see that each color class is an independent set and that the color classes form a partition of V (G).

Deﬁnition 1.2.2 The chromatic number of a graph G, denoted by χ(G), is deﬁned by: χ(G) = min k : G is k-colorable . { }

The chromatic number of a graph is the least number of colors that can be used to proper color the vertices of the graph. This invariant is well-deﬁned, since any graph G can be colored with V (G) colors by assigning each vertex | | a diﬀerent color.

The multiple edges of a graph do not aﬀect the existence of a particular k-coloring or the chromatic number, so for the remainder of the chapter we will assume that all graphs considered are simple.

In any proper coloring of a graph the vertices of a clique receive diﬀerent colors, so we obtain the following lower bound for χ(G):

Remark 1.2.3 χ(G) ω(G). ≥

Even though equality can be attained in the above lower bound (for instance, when the graph G is a clique), there are instances of graphs with ﬁxed clique number and arbitrarily large chromatic number, thus the chromatic

9 number and the clique number can be very far apart. Graphs G which satisfy

χ(H)= ω(H) for any induced subgraph H of G are called perfect graphs.

Deﬁnition 1.2.4 A graph G is called P3-free if it does not contain any induced path of length 3.

The class of P3-free graphs provides a well-known subclass of graphs which are perfect:

Theorem 1.2.5 All P3-free graphs are perfect.

Proof. Let G be a P -free graph. We proceed by induction on n = V (G) . 3 | | The statement of the theorem is obvious for n 3. We may assume therefore ≤ that G has at least 4 vertices, G is connected and that the assertion holds for all graphs with fewer than n vertices. It suﬃces to show that χ(G)= ω(G). If G is a complete graph, then it is obviously perfect. Otherwise, G has a vertex cut-set C and the graph H := G C breaks into connected components \ C ,C , ,Ck with k 2. 1 2 ··· ≥ Any vertex v C is completely joined to H. If this were not the case, ∈ v has at least one non-neighbor in one of the connected components of H, say C1. Since G is connected, it follows that there exists and edge e = xy of G with both endpoints in C1 such that x is not adjacent to v and y is adjacent to v. On the other hand, v has at least one neighbor in C2, say u. The vertices x,y,v,u, in this order, induce a path of length 3, contrary to the hypothesis.

10 We obtain:

χ(G) χ(H)+ χ(C) ω(H)+ ω(C) ω(G), ≤ ≤ ≤ so the graph G is perfect.

The following lemma oﬀers a lower bound for the chromatic number in terms of the size of the graph and its independence number:

Lemma 1.2.6 For any graph G we have:

V (G) χ(G) | |. ≥ α(G)

Proof. Let k = χ(G) and let C ,C , ,Ck be the color classes induced 1 2 ··· by a k-coloring of G. Since the color classes partition V (G), there exists at V (G) least one color class of size greater than or equal to | k |. Any color class is also an independent set, so we obtain

V (G) α(G) | |, ≥ χ(G) which is equivalent to the inequality in the statement of the lemma.

There are instances when the above inequality is very weak, for example when the graph G is a disjoint union of an independent set and a clique of equal sizes. For some of the questions studied later in this paper, it is V (G) possible to reduce the general case to the case when χ(G)=|α(G)|.

11 Graphs with few edges have low chromatic number, as shown by the following:

Proposition 1.2.7 Let e = E(G) be the cardinality of the edge set of the | | graph G. Then: 1 1 χ(G) + + 2e. ≤ 2 r4

Proof. Let k = χ(G) and let C ,C , ,Ck be the color classes induced by 1 2 ··· a k-coloring of G. For any two diﬀerent color classes Ci and Cj, there exists at least one edge between them, otherwise we could combine the two classes k and get a coloring with fewer colors. Therefore G has at least edges. 2 µ ¶ Solving this inequality in terms of k, we obtain the conclusion.

Deﬁnition 1.2.8 A k-edge-coloring of G is a mapping c : E(G) [k] → such that c(e) = c(f) if the edges e and f are adjacent in G. If a graph G 6 admits a k-edge-coloring, we say that G is k-edge-colorable.

Deﬁnition 1.2.9 The chromatic index of a graph G, denoted by χe(G), is deﬁned by:

χe(G) = min k : G is k-edge-colorable . { }

The maximum degree of the graph G, denoted by ∆(G), provides an obvious lower bound for its chromatic index. Unlike in the case of the chromatic number, the distance between these two invariants is always small:

12 Theorem 1.2.10 (Vizing) If G is a simple graph, then:

∆(G) χe(G) ∆(G) + 1. ≤ ≤ We conclude this section with a result about coloring ”almost all” the vertices of a graph with a speciﬁed number of colors. In order to make this statement precise, we need a few deﬁnitions.

Definition 1.2.11 A class of graphs is called closed under disjoint G union if for any two graphs G and H from , not necessarily distinct, their G disjoint union is also a member of . G Definition 1.2.12 Let k be a positive integer. A class of graphs is called G k-colorable if any graph G in is k-colorable. G Definition 1.2.13 Let k be a positive integer and f : N (0, ) be a → ∞ function. We say that a graph G with n = V (G) is (k, f)-colorable if | | there exists an induced k-colorable subgraph of G which has at least n f(n) − vertices.

In other words, a graph is (k, f)-colorable if all except at most f(n) of its vertices can be proper colored with k colors.

Deﬁnition 1.2.14 Let k be a positive integer and f : N (0, ) be a → ∞ function. A class of graphs is called (k, f)-colorable if all its members G are (k, f)-colorable.

Theorem 1.2.15 Let k be a positive integer and f : N (0, ) be a func- → ∞ tion such that lim f(n)/n = 0 as n . Let be a graph class that is → ∞ G closed under disjoint union and is (k, f)-colorable. Then is k-colorable. G 13 Proof. Let G be a member of with n = V (G) and let p be a positive G | | integer with the property that f(np) < p. Such an integer always exists, since f(np)/np 0 as p . → → ∞ Let H be the graph consisting of the disjoint union of p copies of G. Then H and H is (k, f)-colorable. Since V (H) = np, all but at most f(np) ∈ G | | vertices of H are colored using k colors. But f(np) < p, so at least one of the components of H is colored completely, providing thus a k-coloring for G.

As a possible application of the previous theorem, we consider the class of all planar graphs, which is obviously closed under taking disjoint unions.

It is easy to show that planar graphs are 5-colorable. If we could prove that all planar graphs admit a 5-coloring in which one of the color classes is small in size (in the sense of Deﬁnition 1.2.13 for a suitable function f), it would follow that all planar graphs are actually 4-colorable, thus providing a diﬀerent proof for the Four Color Theorem!

1.3 Graph Minors and Hadwiger’s Conjec-

ture

One of the central concepts in Graph Theory is the concept of a minor of a given graph. In order to provide a precise deﬁnition of this concept, we need to describe three common operations performed on graphs:

14 1. vertex deletion: for a graph G and a vertex v V (G), by applying this ∈ operation we obtain a new graph, denoted by G v, given by V (G v)= \ \ V (G) v and E(G v)= e E(G) : e not incident with v in G . If \ { } \ { ∈ } H is a subgraph of G, the subgraph of G obtained by deleting all the vertices in V (H) will be denoted by G H; \

2. edge deletion: for an edge e E(G), deleting the edge e produces a ∈ new graph denoted by G e, deﬁned by V (G e)= V (G) and E(G e)= \ \ \ E(G) e ; \ { }

3. edge contraction: for an edge e = uv, contracting e gives a new graph, denoted by G/e, deﬁned by V (G/e)= V (G) u, v x , G/e retains \ { } ∪ { } all the edges of G not incident to u or v and the new vertex x becomes

adjacent to all the vertices in NG(u) NG(v). ∪

Deﬁnition 1.3.1 Let H and G be two graphs. We say that H is (isomorphic to) a minor of G and we write H M G if H can be obtained from G by ≤ applying the above operations ﬁnitely many times.

An alternative way to state the above deﬁnition is as follows: the graph H with V (H)= v ,v , ,vk is a minor of G if there exist connected vertex- { 1 2 ··· } disjoint subgraphs of G, say C ,C , ,Ck, with the following property: if 1 2 ··· vi and vj are adjacent in H for some 1 i < j k, then there exists an ≤ ≤ edge e E(G) joining Ci and Cj. The subgraphs C ,C , ,Ck are called ∈ 1 2 ··· the nodes of the minor.

15 In the case when H is the complete graph on k vertices, in order to show that H is a minor of G, it suﬃces to exhibit connected vertex-disjoint subgraphs C ,C , ,Ck such that any two of them are joined by an edge. 1 2 ··· By analogy to the clique number of a graph, we provide the following:

Deﬁnition 1.3.2 Let G be a graph. The clique-minor number of G, denoted by ωM (G), is deﬁned by:

ωM (G) = max k : Kk is a minor of G . { }

One of the central open problems in graph coloring is Hadwiger’s Con- jecture, which relates the chromatic number to the size of the largest clique minor contained in a graph:

Conjecture 1.3.3 (Hadwiger) Any graph G which does not contain Kk+1 as a minor is k-colorable.

Using the above notation, the conjecture can be restated as:

Conjecture 1.3.4 For any graph G we have:

χ(G) ωM (G). ≤

Hadwiger’s Conjecture is still open for k 6. For the remainder of this ≥ section, we will take a look at the small cases of the conjecture.

If k = 1, any graph without a K2 minor is edgeless, and therefore 1- colorable. If k = 2, graphs with no K3 minor are precisely the graphs with no cycles (forests). All such graphs are obviously bipartite, so the conjecture holds in this case as well.

16 For k = 3, graphs with no K4 minor form a class called series-parallel graphs. Any such graph can be shown to have a vertex of degree at most 2, therefore we can 3-color all series-parallel graphs by using a greedy algorithm.

The case k = 4 was reduced by Wagner to the Four Color Theorem, which was proved by Appel and Haken and by Robertson, Sanders, Seymour and

Thomas. The case k = 5 was settled by Robertson, Seymour and Thomas, by showing that a minimal counterexample to the conjecture is apex-planar, i.e. planar after the removal of one vertex, thus reducing this case to the Four Color Theorem again.

1.4 Large Clique Minors and Connections with

Other Invariants

In order to attack successfully some special cases of Hadwiger’s Conjecture, it would be very useful to provide bounds for the clique minor number in terms of other graph invariants besides the chromatic number.

17 The obvious candidates are the number of edges (or edge density), the stability number and the usual clique number. Sparse graphs cannot exhibit large clique minors, as shown by the following:

Proposition 1.4.1 For a graph G with e = E(G) we have: | | 1 1 ωM (G) + + 2e. ≤ 2 r4

Proof. Let k = ωM (G). According to a previous remark, we can ﬁnd connected, vertex-disjoint subgraphs of G, C ,C , ,Ck say, such that any 1 2 ··· k(k 1) two of them are joined by an edge. It follows that G has at least 2− edges, and by solving this inequality for e we obtain the conclusion.

Equality can be attained in the above upper bound, for instance when the graph G is complete or edgeless. In other cases, however, the inequality is weak: planar triangulations on n vertices contain no K5, so the clique minor number is bounded, but the number of edges is 3n 6. − The following proposition provides an upper bound for the clique minor number in terms of the clique number:

Proposition 1.4.2 For any graph G we have:

V (G) + ω(G) ωM (G) | | . ≤ 2

Proof. Let n = V (G) , k = ωM (G) and C ,C , ,Ck be vertex-disjoint | | 1 2 ··· connected subgraphs of G, pairwise joined by an edge. Without loss of generality, we may assume that C = C = = Cp = 1 and Ci 2 for | 1| | 2| ··· | | | | ≥ 18 p + 1 i k, for some 0 p k. In other words, we have p nodes of size ≤ ≤ ≤ ≤ one and k p nodes of size at least two, and p may possibly be zero or equal − to k. Since the nodes of the clique minor are vertex-disjoint, we can write:

n Ci 2(k p)+ p = 2k p, ≥ | | ≥ − − i=1 X and therefore: n + p ωM (G) . ≤ 2 Since the nodes of size one form a clique, obviously p ω(G), which gives ≤ the desired conclusion.

The above inequality is useful only if ωM (G) is at least half the size of the graph G itself, otherwise it is trivially true. In some sense, if α(G) = 2, the upper bound in Proposition 1.4.2 is sharp. The α = 2 case will be discussed in detail in the next chapter. Finally, according to Lemma 1.2.6, Hadwiger’s Conjecture implies the following:

Conjecture 1.4.3 For any graph G we have:

V (G) ωM (G) | |. ≥ α(G)

We will prove the following weakening of the above conjecture:

Theorem 1.4.4 Any graph G satisﬁes:

V (G) ωM (G) | | . ≥ 2α(G) 1 −

19 We need a few deﬁnitions and auxiliary results.

Deﬁnition 1.4.5 An induced connected subgraph H of G is called universal if for any v V (G) V (H), there exists an edge between v and a vertex ∈ \ of H. An edge e of G is said to be universal if the subgraph induced by e is universal in G.

It is not hard to see that only the graphs which are connected admit universal subgraphs. On the other hand, Hadwiger’s Conjecture is easily reducible to the connected case. Universal subgraphs provide the main tool for constructing large clique minors, as shown by:

Lemma 1.4.6 If H is a universal subgraph of a graph G, we have:

ωM (G) ωM (G H) + 1. ≥ \

Proof. Let k = ωM (G H) and N ,N , ,Nk be the nodes of a clique \ 1 2 ··· minor of size k in G H. Since H is universal, it is a connected subgraph of \ G and there exists at least one edge between H and each of the subgraphs

N ,N , ,Nk. Therefore, we can obtain a clique minor of size k +1 in G 1 2 ··· with nodes H,N ,N , ,Nk. 1 2 ···

The following lemma shows another way of obtaining clique minors:

Lemma 1.4.7 Let G be a graph and H and K be vertex-disjoint subgraphs of G such that V (H) V (K) = V (G). Assume that for any v V (H) and ∪ ∈ u V (K), u and v are adjacent (we say that H and K are completely ∈ joined). Then:

ωM (G) ωM (H)+ ωM (K). ≥

20 Proof. Let k = ωM (H), p = ωM (K) and M , M , , Mk; N ,N , ,Np 1 2 ··· 1 2 ··· be the nodes of clique minors of sizes k and p in H and K, respectively. We can construct a clique minor of size k + p in the graph G by using the nodes

M , M , , Mk and N ,N , ,Np. 1 2 ··· 1 2 ···

In order to ﬁnd large clique minors, we are interested in constructing universal subgraphs of small size. A good place to start such a construction would be by picking a largest independent set in a graph. Obviously, the vertices in the rest of the graph have at least one neighbor in the independent set. However, such a stable set is not connected if it is of size at least two, so we need additional vertices to build the universal subgraph. The following lemma shows the existence of such a subgraph containing a given vertex:

Lemma 1.4.8 Let G be a connected graph and v be a vertex of G. Then there exists a universal subgraph H of G such that v V (H) and ∈ V (H) 2α(H) 1. | | ≤ −

Proof. We proceed by induction on n = V (G) . The lemma holds for | | n 3. We may assume that n 4 and that the result holds for all graphs ≤ ≥ with fewer vertices than G.

Let F be the graph obtained from G by deleting v and all its neighbors, i.e. F = G (NG(v) v ). If F is the null graph, we can choose H such that \ ∪{ } V (H)= v and the statement of the lemma obviously holds. Therefore, we { } may assume that F is non-null. The graph F may no longer be connected, so it has connected components C ,C , ,Ck, with k 1. 1 2 ··· ≥

21 Since G is connected, there exist edges e ,e , ,ek E(G) such that 1 2 ··· ∈ ei has an endpoint in Ci, say vi, and one endpoint in NG(v), say ui, for all

1 i k. The vertices ui (1 i k) are not necessarily distinct. Applying ≤ ≤ ≤ ≤ the induction hypothesis to the connected graphs C ,C , ,Ck and vertices 1 2 ··· v ,v , ,vk, we deduce the existence of universal subgraphs H ,H , ,Hk 1 2 ··· 1 2 ··· for each of the graphs C ,C , ,Ck. Moreover, we have vi V (Hi) and 1 2 ··· ∈ V (Hi) 2α(Hi) 1 for all 1 i k. | | ≤ − ≤ ≤

Figure 1.1: Constructing a Universal Subgraph.

22 k k Let H be the subgraph of G induced by v ui V (Hi). { }∪ i=1{ }∪ i=1 Then it is easy to check that H is universal for G and:S S

k k

V (H) 1+ k + V (Hi) 1+ k + (2α(Hi) 1) 2α(H) 1, | | ≤ | | ≤ − ≤ − i=1 i=1 X X since:

α(H) 1+ α(H )+ α(H )+ + α(Hk), ≥ 1 2 ··· thus providing the conclusion.

Corollary 1.4.9 Any connected graph G has a universal subgraph with at most 2α(G) 1 vertices. −

We also need the following general result:

Lemma 1.4.10 Let k be a positive integer and a ,a , ,ak and b , b , , bk 1 2 ··· 1 2 ··· be strictly positive reals. Then:

ai a + a + + ak max 1 2 ··· . 1≤i≤k bi ≥ b + b + + bk 1 2 ··· Proof. Without loss of generality, we may assume that

a a 1 = max i , b1 1≤i≤k bi i.e. a bi aib for all 1 i k. By adding these k inequalities side by side, 1 ≥ 1 ≤ ≤ we obtain:

a (b + b + + bk) b (a + a + + ak), 1 1 2 ··· ≥ 1 1 2 ··· which is equivalent to the statement of the lemma.

23 We are now ready to give a proof for Theorem 1.4.4:

Proof. We proceed by induction on n = V (G) . If the graph G is | | not connected, it has connected components C ,C , ,Ck. Let α = α(G) 1 2 ··· and ni = V (Ci) and αi = α(Ci) for 1 i k. Then α + α + + | | ≤ ≤ 1 2 ··· αk = α. The largest clique minor in G is completely contained in one of the components C ,C , ,Ck. Using the induction hypothesis and Lemma 1 2 ··· 1.4.10, we obtain:

ni n ωM (G) max , ≥ 1≤i≤k 2αi 1 ≥ 2α k − − yielding thus the same conclusion for G. We may assume therefore that G is connected. According to Lemma 1.4.8, there exists a universal subgraph H of G of size at most 2α(G) 1. − Let F := G H. Then F has at least V (G) 2α(G) + 1 vertices. Using \ | | − Lemma 1.4.6 and the induction hypothesis for F , we obtain:

V (G) 2α(G) + 1 V (G) ωM (G) ωM (F ) + 1 | | − + 1 | | , ≥ ≥ 2α(F ) 1 ≥ 2α(G) 1 − − showing that the graph G also has the desired property.

24 1.5 Upper Bounds for the Chromatic Num-

ber

The following result provides a lower bound for the clique minor number in terms of the average degree of a graph:

Theorem 1.5.1 (Kostochka 1982; Thomason 1984) There exists a constant c> 0 such that for every positive integer k, every graph of average degree at least ck√log k contains a Kk minor. Up to the value of c, this bound is best possible as a function of k.

The following is an immediate consequence:

Corollary 1.5.2 There exists a constant c > 0 such that any graph G sat- isﬁes:

χ(G) cωM (G) log ωM (G). ≤ p We will provide a new upper bound for the chromatic number of a graph which in some situations is sharper than the one above. We need the following:

Lemma 1.5.3 Any graph G admits an induced subgraph H such that V (G) χ(H) ωM (G) and V (H) | |. ≤ | | ≥ 2

Proof. We proceed by induction on n = V (G) . We may assume that | | the result holds for n 3 and for all graphs with fewer than n vertices. ≤ If G is not connected, we can use the induction hypothesis for each of its connected components to obtain the conclusion.

25 By applying Lemma 1.4.8, we deduce the existence of a universal subgraph

U such that V (U) 2α(U) 1. In other words, the induced subgraph U | | ≤ − contains an independent set, A say, larger than half its number of vertices.

From the induction hypothesis, the graph G U has an induced subgraph H′ \ V (G U) ′ of size at least | \ | and with χ(H ) ωM (G U). 2 ≤ \ Let H be the subgraph of G induced by A V (H′). Then, since A is ∪ stable, we have:

′ χ(H) 1+ χ(H ) 1+ ωM (G U) ωM (G), ≤ ≤ \ ≤ since U is universal. On the other hand:

V (U) V (G U) V (G) V (H) A + V (H′) | | + | \ | = | |, | | ≥ | | | | ≥ 2 2 2 which shows that the statement holds for G.

Theorem 1.5.4 For any graph G with n = V (G) and ωM = ωM (G) we | | have: n χ(G) ωM log + 1 . ≤ 2 ω µ» M ¼ ¶

Proof. From the previous lemma, we have the existence of a ωM - colorable induced subgraph of G, say H1, such that:

n V (G H ) . | \ 1 | ≤ 2

Let G = G H , by applying the lemma again for G , we obtain a new 2 \ 1 2 induced subgraph H with χ(H ) ωM (G ) ωM and such that G H 2 2 ≤ 2 ≤ 2 \ 2 n has at most vertices. 4

26 Continuing this process, after p steps we have constructed vertex-disjoint subgraphs H ,H , ,Hp, each of them ωM -colorable, such that: 1 2 ··· p n V (G Hi) . | \ | ≤ 2p i=1 [ This implies that: n χ(G) pωM + . ≤ 2p n n If we choose p = log , it follows that ωM and we obtain the 2 ω 2p ≤ » M ¼ conclusion.

For graphs with bounded stability number, we obtain the following linear upper bound for the chromatic number in terms of the clique minor number:

Corollary 1.5.5 For any graph G with n = V (G) , ωM = ωM (G) and | | α = α(G) we have:

χ(G) ωM ( log α + 2) . ≤ ⌈ 2 ⌉ n Proof. From Theorem 1.4.4, we obtain that 2α. This inequality, ωM ≤ together with the previous theorem, imply the desired result.

27 CHAPTER 2 HADWIGER’S CONJECTURE FOR GRAPHS WITH STABILITY NUMBER 2

2.1 Preliminaries

It was stated earlier that Hadwiger’s Conjecture is known for k 5, i.e. for ≤ graphs which do not contain K6 as a minor. This approach attempts to prove the conjecture by bounding the clique minor number of the graph.

An alternative approach would be to bound (or ﬁx) other invariants of the graph and to try to show the desired upper bound for the chromatic

28 number. The obvious candidates for this are the clique number and the stability number. Bounding the clique number seems to lead to extremely diﬃcult problems, since even triangle-free graphs are not known to satisfy

Hadwiger’s Conjecture. When attempting to bound the stability number, the ﬁrst interesting case arises very early, for α(G) = 2.

Indeed, the only graphs for which α(G) = 1 are the complete ones, and the conjecture here is trivial. If α(G) = 2, Conjecture 1.4.3, which has to hold if Hadwiger’s Conjecture holds, becomes:

Conjecture 2.1.1 For any graph G with α(G) = 2 we have:

n ωM (G) , ≥ 2 where n = V (G) . | |

It can be shown that for the α = 2 case, the above conjecture and Had- wiger’s Conjecture are, in fact, equivalent (see Theorem 2.1.4). On the other hand, it is believed that if Hadwiger’s Conjecture fails, the ﬁrst counterex- amples should arise in this case, since we have to be able to construct such a large clique minor (half the size of the graph itself) in all situations.

Deﬁnition 2.1.2 The complement of a graph G, denoted by G, is a graph with the same vertex-set as G, in which two vertices are adjacent if and only if they are not adjacent in G.

Deﬁnition 2.1.3 A class of graphs is called closed under induced sub- G graphs if for any G , any induced subgraph of G is also a member of ∈ G . G 29 Theorem 2.1.4 Let be a graph class closed under induced subgraphs. As- G V (G) sume that ωM (G) | | for any G . Then any graph of satisﬁes ≥ 2 ∈ G G Hadwiger’s Conjecture.

V (G) Proof. Let G such that ωM (G) | |. Assume that the theorem ∈ G ≥ 2 holds for all graphs with fewer vertices than G. If the complement of G has a perfect matching, then we can take the edges of the matching as color classes V (G) in G, so χ(G) | |, thus G satisﬁes Hadwiger’s Conjecture. ≤ 2 Otherwise, according to Tutte’s Matching Theorem and Theorem 1.1.13, there exists a set S such that G S has odd components C ,C , ,Cp with \ 1 2 ··· S

ωM (G) ωM (Ci)+ ωM (Di). ≥ i=1 i=1 X X In any coloring of G S, the vertices of two diﬀerent components of G S \ \ receive diﬀerent colors.

V (Ci) + 1 We can color each C with | | colors such that one color, say c , i 2 i appears only at vi, for all 1 i p. Then we can use colors c ,c , ,cm ≤ ≤ 1 2 ···

30 to color vertices u , u , , um, so it follows that χ(G)= χ(G S). Since: 1 2 ··· \ k p k p

χ(G S) χ(Ci)+ χ(Di) ωM (Ci)+ ωM (Di) ωM (G), \ ≤ ≤ ≤ i=1 i=1 i=1 i=1 X X X X we obtain the desired conclusion.

According to classical Ramsey Theory, any graph satisfying α(G) = 2 contains a clique of size at least (√n log n) and examples have been con- O structed with clique numbers of this order of magnitude. Proposition 1.4.2, together with Hadwiger’s Conjecture would imply that there exist graphs n having clique minor numbers of size + (√n log n), essentially of the same 2 O order of magnitude as the lower bound of Conjecture 2.1.1. The rest of this chapter is dedicated to proving Hadwiger’s Conjecture for the α = 2 case with some additional restrictions. Excluding certain graphs as induced subgraphs or imposing edge-density conditions seems to be the only way to make progress towards proving the general case.

2.2 Lower Bounds for the α = 2 Case

In a graph G, a seagull is an induced path of length 2. The divalent vertex of the path is called the center of the seagull.

For v V (G), let NG(v) be the set of all vertices of G adjacent to v. The ∈ subscript G may be omitted if there is no possibility of confusion. A simple graph which does not contain an induced path of length 2 is called seagull-free. The structure of such graphs is given by the following:

31 Lemma 2.2.1 If G is a seagull-free graph, then each connected component of G is a clique.

Proof. Without loss of generality we may assume that G is connected. If

G is not a clique, then there exist two non-adjacent vertices u and v, say. Since G is connected, there exists a shortest induced u v path P which − has length at least 2, a contradiction to the fact that the graph G is seagull- free.

The following two remarks are obvious:

Remark 2.2.2 If α(G) = 2 and v V (G), then V (G) (N(v) v ) is a ∈ \ ∪ { } clique.

Remark 2.2.3 If α(G) = 2 and H is a seagull, then H is universal.

By using seagulls as the nodes of the clique minor, we obtain the following:

Proposition 2.2.4 Let G be a graph with α(G) = 2. Then:

V (G) ωM (G) | |. ≥ 3

Proof. Induction on n = V (G) . For n = 1, 2, 3, the statement is | | obvious. Assume n 4 and that the proposition is true for all graphs with ≥ fewer than n vertices.

32 If G contains a seagull H, then by lemma 1.4.6 we have:

n 3 n ωM (G) ωM (G H) + 1 − +1= . ≥ \ ≥ 3 3

Otherwise G is seagull-free and each of its connected components is a clique. Since α(G) = 2, G has at most two such components, so we obtain n ω(G) . ≥ 2

With a little bit of additional work, it is possible to prove the following strengthening of the previous proposition:

Proposition 2.2.5 Let G be a graph with α(G) = 2. Then:

V (G) + ω(G) ωM (G) | | . ≥ 3

Proof. Let n = V (G) and K be a clique of G of size ω(G). Let | | S ,S , ,Sk be vertex-disjoint seagulls of G, none of which having any 1 2 ··· k vertex in K and such that H := G K Si is seagull-free. In other \ ∪ i=1 words, we successively remove seagulls³ in G SK until´ this is no longer possible; \ the resulting graph H is seagull-free and according to Lemma 2.2.1, is either a clique or a disjoint union of two cliques.

If H is a clique, then V (H) + V (K) 2ω(G) and therefore | | | | ≤ n 2ω(G) k − . Since any seagull is universal, by using the vertices of K ≥ 3 and S ,S , ,Sk as nodes, we obtain that: 1 2 ··· n 2ω(G) n + ω(G) ωM (G) − + ω(G)= . ≥ 3 3

Otherwise, H is a disjoint union of two cliques K1 and K2, say. Any vertex in K is completely joined to at least one of the cliques K1 and K2, otherwise G

33 contains an independent set of size three. It follows that H K is expressible ∪ as a union of two cliques as well, and therefore we have V (H) + V (K) | | | | ≤ 2ω(G) again. Proceeding as in the previous case, we complete the proof of the proposition.

Even though it seems that the last proposition provides a signiﬁcant improvement over Proposition 2.2.4, as it was noted before in this paper, there n are graphs with α = 2 having the largest clique of size (√n log n). The O 3 is still essentially the best lower bound known for the general α = 2 case.

In order to obtain an improvement of order (n), we will impose additional O density conditions in Section 2.5.

2.3 Graphs with Low Connectivity

The purpose of this section is to prove that any graph G with α(G) = 2 containing a vertex cut-set of size at most half the number of the vertices of the graph has the desired clique minor.

Theorem 2.3.1 Let G be a graph with α(G) = 2. Let k = k(G) be the connectivity of G. Then:

n ωM (G) min n k, , ≥ − 2 ³ ´ where n = V (G) . | |

Proof. We may assume that G is connected, otherwise k = 0 and the statement is obviously true. Let C be a cut-set of G such that C = k. Since | | 34 α(G) = 2, G C is a disjoint union of two cliques, say K and K . Moreover, \ 1 2 any vertex of C is completely joined with at least one of the cliques K1 and

K2.

Let C1 be the set of all the vertices in C completely joined to K1 and let C := C C . Any vertex of C completely joined to the clique K . 2 \ 1 2 2

Let m1 = min(C1,K2) and m2 = min(C2,K1). Deﬁne the bipartite graph

B1 = (V1, V2) as follows: V1 = C1, V2 = K2 and E(B1) consists of all the edges of G with one endpoint in C1 and the other in K2. The bipartite graph

B2 is deﬁned in a similar fashion on (K1,C2).

We claim that B1 contains a matching of size m1. Otherwise, by K¨onig’s Theorem, there exists a cover S for the edges of B such that S

ωM (G) m + m = K + K = n k. ≥ 1 2 | 1| | 2| −

Case 2: C < K , C K . | 1| | 2| | 2| ≥ | 1| Then m = K and by considering the edges of M and the vertices of 2 | 1| 2

K2 as nodes, we get:

ωM (G) m + K = n k. ≥ 2 | 2| − 35 Case 3: C < K , C < K . | 1| | 2| | 2| | 1| Without loss of generality we may assume that K C K C . | 1| − | 2| ≥ | 2| − | 1| We can take the edges of M plus the vertices of K C as the nodes of the 1 \ 2 minor; thus:

n ωM (G) M + K C = C + C + K C = K + C , ≥ | | | 1| − | 2| | 1| | 2| | 1| − | 2| | 1| | 1| ≥ 2

and the proof is complete.

Corollary 2.3.2 If G is a graph with α(G) = 2 and connectivity at most V (G) | |, then: 2 V (G) ωM (G) | |. ≥ 2

Proof. The result follows directly from the previous theorem.

2.4 Excluding Certain Subgraphs

The main diﬃculty in proving Hadwiger’s Conjecture in the α = 2 case lies in the lack of an adequate structure theorem for such graphs. Indeed, graphs with stability number equal to two are exactly the complements of triangle- free graphs. These graphs have been studied extensively in the past, but no general precise structure seems to be known for them.

By excluding certain graphs as induced subgraphs, such a structure theory becomes possible. In this section, we will show that C4 or C5 free graphs (i.e. graphs not containing a C4 or a C5 as an induced subgraph) with α = 2

36 satisfy Hadwiger’s Conjecture. Since we need to construct a clique minor of size half the number of the vertices of the graph, we expect most of the nodes in the minor to be of size one or two. The existence of a universal edge in such a graph would prove to be very useful for this purpose. Unfortunately, not all graphs with stability number two admit a universal edge, as the example of C5 itself easily shows. Excluding C5 as an induced subgraph will guarantee the existence of such an edge, as we will see later this section.

The following lemma deals with the existence of a cut-vertex in the graph G:

Lemma 2.4.1 Let G be a connected graph with α(G) = 2 and let v be a cut-vertex of G. Then there exists a universal edge e incident with v.

Proof. Since α(G v) = 2 and G v is disconnected, it follows that \ \ G v is the disjoint union of exactly two cliques K and K with no edges in \ 1 2 between. Moreover, v is completely joined to at least one of the cliques K1 and K2. We may assume that v is complete to K1. Since G is connected, there exists a vertex u V (K ) N(v). It is easy now to check that the edge ∈ 2 ∩ e = uv is universal in G.

We can now show that a C5-free graph with α = 2 has a universal edge meeting a ﬁxed maximal clique:

Lemma 2.4.2 Let G be a connected graph with α(G) = 2. Let K be a maximal clique in G and let v V (K). Then at least one of the following ∈ statements holds:

37 1. v is a cut-vertex of G;

2. there exists an induced cycle of length ﬁve containing a vertex of K;

3. there exists a vertex u V (K), u = v and there exists a universal edge ∈ 6 e incident with u.

Proof. Induction on n = V (G) . We may assume that n 4 and that | | ≥ the statement holds for all graphs with fewer than n vertices.

Case 1: K v is a maximal clique in G v. \ \ Let G′ = G v and K′ = K v. We may assume that G′ is connected, \ \ otherwise (1) holds. Let v′ K′ and apply the induction hypothesis for ∈ G′,K′ and v′.

If (1) holds for G′, i.e. v′ is a cut-vertex of G′, then by Lemma 2.4.1 there exists an edge e incident to v′ which is universal in G′. But e is also universal in G, so (3) holds for G.

If (2) holds for G′, then (2) holds for G. If (3) holds for G′,K′ and v′, then (3) holds for G.

Case 2: K v is not maximal in G v. \ \ Then there exists a vertex b in G K such that b is adjacent to all the \ vertices of K except v. Let G′′ = G b. If b is a cut-vertex of G, then again \ G b = K K , with K and K disjoint cliques. We may assume that b is \ 1 ∪ 2 1 2 completely joined to K ; then v K . Choose any u K v, then u K 1 ∈ 2 ∈ \ ∈ 2 and the edge ub is universal in G, so (3) holds. Therefore we may assume that G′′ is connected. Apply the induction hypothesis for G′′,K and v.

38 If (2) holds for G′′, then (2) holds for G.

If (3) holds for G′′, then (3) holds for G. If (1) holds for G′′, then v is a cut-vertex in G′′, so G′′ v = K K and \ 1 ∪ 2 v is completely joined to K1, say. Case 2.1: K v is contained in K . \ 2 Choose any u K v, then e = uv is universal in G. ∈ \ Case 2.2: K v = K . \ 1 Then b is complete to K and if v is completely joined to K , any u K 1 2 ∈ 1 gives a universal edge e = uv for G. So we may assume that there exists a

′ ′ vertex v in K2 such that v and v are not adjacent. If b is completely joined to K , choose any u K and the edge ub is 2 ∈ 1 universal in G. Otherwise, there exists a vertex b′ K such that b and b′ ∈ 2 are not joined. It follows that b′ = v′, v is joined to b′ and b is joined to 6 v′. For any u K , the vertices u, v, b′,v′, b form an induced 5-cycle passing ∈ 1 through K, so (2) holds for G.

We are now in a position to show that the desired lower bound for the clique minor number holds for C5-free graphs:

Theorem 2.4.3 Let G be a C5-free graph with α(G) = 2. Then:

n ωM (G) , ≥ 2 where n = V (G) . | |

Proof. Induction on n. Obvious for n = 1, 2, 3. If G is not connected, then G = K K , with K ,K cliques, so the statement holds. Therefore 1 ∪ 2 1 2 39 we may assume that G is connected. Let K be a maximal clique in G and let v K. Apply Lemma 2.4.2; either v is a cut-vertex of G and by Lemma ∈ 2.4.1, G has a universal edge e incident to v, or there exists u K and a ∈ universal edge e incident with u. Let H be the subgraph obtained from G by deleting the endpoints of e. We have:

n 2 n ωM (G) ωM (H) + 1 − +1= . ≥ ≥ 2 2

Corollary 2.4.4 If G is a C5-free graph with α(G) = 2, then Hadwiger’s Conjecture holds for G.

Proof. This follows directly from Theorem 2.4.3 via the standard reduc- tion argument from Theorem 2.1.4.

We now turn our attention to the C4-free case. Graphs with α = 2, with no induced C4, but containing an induced C5 have a nice structure, as shown in the proof of the following:

Theorem 2.4.5 Let G be a C4-free graph with α(G) = 2. Then:

n ωM (G) , ≥ 2 where n = V (G) . | |

Proof. Without loss of generality we may assume that G is connected and G contains an induced cycle of length ﬁve. Let C = v1v2v3v4v5 be such a cycle.

40 Since α(G) = 2, any vertex v G C is adjacent to either 3, 4 or 5 ∈ \ consecutive vertices of C. Let Ai be the set of all vertices in G C joined \ in C only to vi,vi ,vi (including vi itself) for i 1, 2, , 5 , where +1 +2 +1 ∈ { ··· } v6 = v1, v7 = v2.

Figure 2.1: The C4-free Case.

Let Ki = Ai Ai for i 1, 2, , 5 , where A = A . It is easy to ∪ +1 ∈ { ··· } 6 1 see that each Ki is a clique in G. Let H be the subgraph of G induced by

5 m Ai and let m = V (H) . We will show that ωM (H) . i=1 | | ≥ 2 S 41 Let k = A and assume that Ai k, for any i 2, 3, 4, 5 . Then: | 1| | | ≥ ∈ { }

K + K = m k, | 2| | 4| − m k so we may assume that K − . We can obtain a clique minor of size | 4| ≥ 2 m for H by using as nodes in the minor the vertices of K and k disjoint 2 4 induced subgraphs G ,G , ,Gk, each with 3 vertices, one in A , one in A 1 2 ··· 1 2 and one in A3. If G = H, the statement of the theorem holds. Otherwise, let B = V (G) V (H) and assume that B is non-empty. \ Each vertex of B is joined to exactly 4 or 5 vertices of C. Since G is

C4-free, no vertex in B can be adjacent to only 4 vertices in C, so it remains that B is completely joined to C. Now let v H C and u B. We may ∈ − ∈ assume that v A . If u and v are not adjacent, then v ,v,v and u form ∈ 1 1 3 an induced C4 in G, contradiction. Denote by K the subgraph of G induced by B. It follows that K is completely joined to H. n m Then ωM (K) − by the induction hypothesis, and by Lemma 1.2 ≥ 2 we obtain:

m n m n ωM (G) ωM (H)+ ωM (K) + − = , ≥ ≥ 2 2 2 so the statement of the theorem holds for G.

42 Corollary 2.4.6 Hadwiger’s Conjecture holds for C4-free graphs with stability number 2.

Proof. It follows directly from Theorem 2.4.5 and Theorem 2.1.4.

2.5 Graphs with High Edge Density

For the remainder of this chapter, we will study graphs with α = 2 which have high edge density. This concept will be made precise later on in this section, when we will state a theorem providing an improvement of order

(n) over the lower bound given by Proposition 2.2.4. O First, we give a few deﬁnitions and auxiliary results:

Deﬁnition 2.5.1 Let G be a graph. A universal matching in G is a matching M with the following property: for any two distinct edges e and e′ of M, there exists at least one edge f E(G) adjacent to e and e′. ∈

In other words, any two distinct edges of a universal matching are connected by at least one other edge of the graph. By contracting each edge of such a matching we obtain a clique minor in G, so by analogy with the clique minor number, we introduce the following:

Deﬁnition 2.5.2 The 2-clique-minor number of a graph G, denoted by

ω2(G), is deﬁned as:

ω (G) = max M : M is a universal matching in G . 2 {| | }

43 The invariant ω2(G) deﬁned above is the size of the largest clique minor of the graph G in which all the nodes have size exactly two. Obviously

ωM (G) ω (G). ≥ 2 We have the following upper bound for the clique minor number:

Proposition 2.5.3 For any graph G we have:

V (G) + ω2(G) + 2ω(G) ωM (G) | | . ≤ 3

Proof. Let n = V (G) , k = ωM (G) and N ,N , ,Nk be the nodes of | | 1 2 ··· a largest clique minor of G. Without loss of generality, we may assume that

N ,N , ,Ns are nodes of size one, Ns , ,Ns t are nodes of size two, 1 2 ··· +1 ··· + and the remaining nodes are of size at least three. Thus our clique minor has s nodes of size one, t nodes of size two and k s t nodes of size three − − or more. Since N ,N , ,Nk are vertex-disjoint, we have: 1 2 ··· k

n Ni 3(k s t) + 2t + s = 3k t 2s. ≥ | | ≥ − − − − i=1 X By solving for k in the above inequality and taking into account that s ω(G) and t ω (G), we obtain the conclusion. ≤ ≤ 2

The above proposition is especially relevant for the α = 2 case. If Hadwiger’s Conjecture holds for this case, since most of these graphs have n ω(G)= (√n log n), it follows that ω (G) has to be very close to . There- O 2 2 fore, for most of the graphs with α = 2, the largest clique minor arises from a universal matching.

44 Even if the clique number can be as small as (√n log n), the 2-clique O minor number has to be a signiﬁcant fraction of the number of vertices, as shown in the following:

Theorem 2.5.4 If Hadwiger’s Conjecture holds for all graphs with stability number two, then any such graph satisﬁes:

V (G) ω (G) | |. 2 ≥ 6 n Proof. Let n = V (G) and assume that ωM (G) . We want to | | ≥ 2 investigate under what conditions we have ω(G) ω (G). ≤ 2 Let K be a clique of G of size ω(G). If G = K, there is nothing left to prove. Otherwise, we construct a bipartite graph B =(V1, V2) deﬁned as follows: V = V (K), V = V (G) V (K) and E(B) consists of all the edges 1 2 \ of G having exactly one endpoint in K. If B has a matching M of size V (K) , we can use M to obtain a universal | | matching in G of size ω(G), so in this case we have ω(G) ω (G). ≤ 2 By K¨onig’s Theorem, if the largest matching in B has size strictly less than K, there exists a cover for the edges of B, say C, such that C <ω(G). | | Let K = K C and K =(G K) C. It follows that there are no edges of 1 \ 2 \ \

G from K1 to K2, and since α(G) = 2, K1 and K2 are cliques of G. Without loss of generality we may assume that K K . We obtain: | 1| ≥ | 2|

n C n ω(G) + 1 ω(G) K − | | − , ≥ | 1| ≥ 2 ≥ 2 which yields: n + 1 ω(G) . ≥ 3 45 n+1 We have shown that either ω(G) ω (G) or ω(G) . In the ﬁrst ≤ 2 ≥ 3 case, by using the previous proposition and the assumption about ωM (G), n+1 we obtain the conclusion. In the second case, a clique of size at least 3 gives immediately a universal matching of the desired size, completing the proof of the theorem.

From the above discussion, it is clear that in order to prove Hadwiger’s

Conjecture for α = 2 (or even only to get an improvement to the bound in Proposition 2.2.4), we have to prove the existence of a universal matching of size c V (G) , where c > 0 is a global constant. On the other hand, a | | universal matching of signiﬁcant size would indeed provide an improvement in Proposition 2.2.4, as shown by the following:

Lemma 2.5.5 Any graph G with α(G) = 2 satisﬁes:

V (G) ω2(G) ωM (G) | | + . ≥ 3 9

Proof. Let n = V (G) and let M be a universal matching of size ω (G). | | 2 Let S ,S , ,Sk be vertex-disjoint seagulls not meeting any edge of M 1 2 ··· k with the property that the graph H := G M Si is seagull-free. \ ∪ i=1 The graph H is obtained from G M by removing³ S seagulls´ until no longer \ possible. Then H is either a clique or a disjoint union of two cliques, so V (H) 2ω(G). Therefore we have: | | ≤ n 2 M 2ω(G) k − | | − . ≥ 3

46 The seagulls S ,S , ,Sk and the edges of M form the nodes of a clique 1 2 ··· minor in G, so:

n 2 M 2ω(G) n + ω2(G) 2ω(G) ωM (G) k + M − | | − + M = − . ≥ | | ≥ 3 | | 3

On the other hand, Proposition 2.2.5 implies that:

2n + 2ω(G) 2ωM (G) . ≥ 3

From the addition of the last two inequalities side by side, we obtain the conclusion.

For any graph G, denote by d(G) its average degree. The following lemma provides a lower bound for the clique number of a graph in terms of its average degree and will prove useful in constructing large clique minors:

Lemma 2.5.6 For any graph G we have:

n ω(G) , ≥ n d(G) − where n = V (G) . | |

Proof. Let e = E(G) and let r be a positive integer. Tur´an’s Theorem | | implies that if: 1 r 2 e> n2 − , 2 r 1 − then Kr is a subgraph of G, i.e. ω(G) r. ≥

47 Taking into account that 2e = nd(G) and solving the above inequality for r, we obtain: n + 1 > r. n d(G) − n By letting r be the smallest integer greater than or equal to n d(G), the − last inequality is satisﬁed, which guarantees the existence of a clique of size r, completing thus the proof of the lemma.

For any graph G, construct a new graph, denoted by G, deﬁned as follows:

V (G)= E(G) and two edges of G are adjacent in G if theye form a 2-matching in Ge and are joined by at least one other edge.e If the graph G contains a large clique, then G has a large clique minor, as shown in the following:e

Lemma 2.5.7 Any graph G satisﬁes:

ω (G) ω(G). 2 ≥ e Proof. Let K be a clique of G of size ω(G). It is easy to check that the vertices of K form a universal matchinge in Geof size ω(G). e The main idea will be to use Lemma 2.5.6 for the graph G to obtain a large clique, providing therefore a large clique minor in G itself.e We expect this approach to work only if the graph G is dense enough. If α(G) = 2, then

G is obviously dense and this propertye should translate to G as well. The following lemma provides the connection between d(G) and de(G): e

48 Lemma 2.5.8 Let c> 1 be a constant. Then there exists a constant c′ > 0, depending on c, such that for any graph G with α(G) = 2 and ∆(G) cδ(G), ≤ we have: (n 1 d(G))4 d(G) e 2n c′ − − , ≥ − − (d(G))2 where n = V (G) and e = E(G) . | | | |

Proof. In order estimate the average degree of G, we construct the following bipartite graph: B = (V1, V2), given by V1 =e V (G), V2 = E(G) and v V is adjacent to e V if the edge e has both endpoints in G ∈ 1 ∈ 2 \ ( v NG(v)), i.e. if v misses e. { }∪ A vertex of G misses, on average, n 1 d(G) vertices. Since α(G) = − − 2, the non-neighborhood of a vertex is a clique, so a vertex of G misses 2 2 (n 1 d(G)) n(n 1 d(G)) − −2 edges of G. This in turn implies that B has − −2 edges. Since V = e, it follows that each edge of G misses, on average | 2| n(n 1 d(G))2 (n 1 d(G))2 − − = − − 2e d(G) vertices of G. But the non-neighborhood of any edge is also a clique, so any 4 (n 1 d(G)) edge of G misses, on average, − − edges of G. 2(d(G))2 Two edges meeting at a vertex are not adjacent in G. Since there are at most 2(n 1) edges incident to the endpoints of any given edge of G, we − e deduce that any edge of G, which is, in turn, a vertex of G, has, on average

4 ′ (n 1 d(G))e V (G) 1 2(n 1) c − − | | − − − − (d(G))2 e neighbors in G, where c′ is a suitable positive constant.

e 49 The condition ∆(G) cδ(G) in the hypothesis of the lemma is needed ≤ to perform the transition between the average number of vertices missed by a vertex and the average number of edges missed by a vertex. The same problem arises when averaging for edges. This is also the reason for the need of the constant c′ in the conclusion.

The following result shows that if the graph G is dense enough, we can obtain a linear improvement in the bound from Proposition 2.2.4:

Theorem 2.5.9 Let c > 0 be a constant. Then there exists a constant c′ > 0, depending on c, with the following property: any graph G with α(G) = 2

3 and d(G) n cn 4 satisﬁes: ≥ −

n ′ ωM (G) + c n, ≥ 3 where n = V (G) . | |

Proof. If G satisﬁes ∆(G) cδ(G), we can apply the previous lemma; ≤ using the additional condition on d(G) yields:

n3 d(G) e 2n a e bn, ≥ − − (d(G))2 ≥ − e for suitable constants a,b > 0.

From Lemma 2.5.6 applied for G, we obtain:

e e nd(G) ′ ω(G) = b n, ≥ bn 2bn ≥ e for some constant b′ > 0.

50 Combining lemmas 2.5.5 and 2.5.7 gives:

′ n ω2(G) n b n ωM (G) + + , ≥ 3 9 ≥ 3 9 b′ so we can choose c′ = . 9 Otherwise, ∆(G) > cδ(G), so there exists a vertex of G of degree at most n c . Since the non-neighborhood of any vertex is a clique, we deduce that: n ω(G) n 1 b′′n, ≥ − c − ≥ for some b′′ > 0. Finally, we have:

n + ω(G) n + b′′n ωM (G) , ≥ 3 ≥ 3 b′′ so we can choose c′ = . Also note that all the constants a, b, b′, b′′ and c′ 3 depend on c only, not on the graph G or on n = V (G) . | |

If we drop the density condition for G, we obtain the following improvement of Proposition 2.2.5:

Theorem 2.5.10 There exists a global constant c > 0 such that any graph

G with α(G) = 2 and n = V (G) satisﬁes: | |

n 4 5 ωM (G) + cn . ≥ 3

51 Proof. Let a> 0 be a constant.

4 If the graph G satisﬁes d(G) < n an 5 , then there exists a vertex − 4 v V (G) such that its non-neighborhood contains at least an 5 vertices, ∈ 4 so ω(G) an 5 . We can now use Proposition 2.2.5 to obtain: ≥ 4 n + an 5 ωM (G) . ≥ 3

4 Otherwise, d(G) n an 5 and we can proceed similarly to the proof of ≥ − the previous theorem. The graph G has a clique of size

e e 4 e ′ 5 ω(G) 6 b n , ≥ e d(G) ≥ bn 5 ≥ − e′ for suitable constants b, b > 0, so wee can invoke Lemma 2.5.5 to get:

′ 4 n ω2(G) n b n 5 ωM (G) + + . ≥ 3 9 ≥ 3 9

Note that the constants b, b′ > 0 depend only on the constant a chosen initially, so we can choose

a b′ c := min , , 3 9 µ ¶ completing thus the proof.

52 CHAPTER 3 GRAPH COMPLEMENTS

3.1 Preliminaries

The main purpose of this chapter is to study how the graph invariants considered so far (such as the clique minor number or the chromatic number) of a graph relate to those of its complement.

The main result of this chapter states that the product of the clique minor numbers of a graph and its complement is at least the number of the vertices of the graph. It is easy to see that this result is implied by

Hadwiger’s Conjecture, so the question whether the above statement holds is very natural.

A fairly surprising fact, which will be shown later in this section, is that if we bound the clique minor number of the complement of a graph, the graph itself contains a clique minor almost as large as half the number of vertices.

53 By excluding relatively small graphs, either as induced subgraphs or as minors, from the complement of a graph forces it to be fairly dense, and we can hope that Hadwiger’s Conjecture becomes tractable in this case. We will use this approach for complements of graphs with no induced C4 or with no

K4 minor. We start with a few deﬁnitions:

Deﬁnition 3.1.1 For a graph G and x, y V (G), the distance between x ∈ and y, denoted by dG(x, y), is the length of the shortest path linking x and y if the two vertices lie in the same connected component of G and otherwise. ∞

Deﬁnition 3.1.2 The diameter of a graph G, denoted by dia(G), is given by:

dia(G) = max dG(x, y). x,y∈V (G)

The main result of this chapter is the following:

Theorem 3.1.3 There exists n N such that any simple graph G with 0 ∈ n = V (G) n satisﬁes: | | ≥ 0

ωM (G)ωM (G) n. ≥

Remark 3.1.4 For any graph G we have α(G)= ω(G) ωM (G), so Theo- ≤ rem 3.1.3 would be a direct consequence of Conjecture 1.4.3, which is in turn implied by Hadwiger’s Conjecture.

54 The following lemma shows that complements of graphs with diameter at least three have a universal edge:

Lemma 3.1.5 For any graph G, either G has a universal edge or dia(G) 2. ≤

Proof. Assume that dia(G) 3, so there exist vertices x and y such that ≥ d (x, y) 3. Then x and y are adjacent in G and the edge xy is universal G ≥ in G.

Theorem 1.5.1 implies that there exists a global constant c> 0 such that any graph G with δ(G) c k√log k satisﬁes ωM (G) k. We can therefore ≥ ≥ deﬁne a function f : N∗ N∗ given by: →

f(k)= max δ(G) : ωM (G)= k . { }

In other words, for a given positive integer k, f(k) is the largest possible minimum degree of a graph G such that Kk+1 is not a minor of G. Also we have:

Remark 3.1.6 There exists a constant c such that the function f deﬁned above satisﬁes:

f(k) c k log k. ≤ p Based on the function f deﬁned previously, we can deﬁne a new function

F : N∗ N∗ given by: → n 1 F (n)= f(i). 2 i=1 X

55 We have the following upper bound for F :

Remark 3.1.7 c F (n) n2 log n, ≤ 2 p where c is the same constant as in Remark 3.1.6.

The bounds for f and F provided above are far from being sharp for small values of their argument. The following theorems are due to Mader and will allow the computation of exact values for the small cases:

Theorem 3.1.8 Every graph G without a K4 minor has a vertex of degree at most two.

Theorem 3.1.9 Every graph with n vertices and more than 3n 6 edges − contains a K5 minor.

Theorem 3.1.10 Every graph with n vertices and more than 4n 10 edges − contains a K6 minor.

The values of f(n) and F (n) for n 5 are summarized in Figure 3.1. ≤

The following lemma provides a lower bound for the sum of the clique minor numbers of a graph and its complement:

Lemma 3.1.11 For any graph G we have:

n ωM (G)+ ωM (G) a , ≥ √log n where n = V (G) and a> 0 is a global constant. | | 56 n f(n) F (n)

1 0 0

2 1 1/2

3 2 3/2

4 5 4

5 7 15/2

Figure 3.1: The values of f and F for n 5. ≤

Proof. Let k = ωM (G) and p = ωM (G). From Theorem 1.5.1, we deduce that d(G)

We are now ready to state the main result of this section:

Theorem 3.1.12 Let G be a graph with n = V (G) , ωM = ωM (G) and | | ωM = ωM (G). Then: n ωM F (ωM ). ≥ 2 −

57 Proof. We proceed by induction on n, the number of vertices of the graph G. We may assume that n 4 and that the statement holds for all ≥ graphs with fewer than n vertices.

According to Lemma 3.1.5, either G has a universal edge e or dia(G) 2. ≤ If e is universal in G, let H be the subgraph obtained from G by deleting the endpoints of e. By applying the induction hypothesis for H, noting also that ωM (H) ωM , we have: ≤ n 2 n ωM ωM (H) + 1 − F (ωM (H))+1 F (ωM ), ≥ ≥ 2 − ≥ 2 − so the statement holds for G also. Otherwise, G has diameter at most two. Let v V (G) such that d (v)= ∈ G δ(G) and let S be the subgraph of G induced by v N (v). Now it is easy { }∪ G to see that S is universal in G and d (v) f(ωM ). G ≤ Let H = G S and let m = V (H) . Then H is an induced subgraph \ | | of G and m n f(ωM ) 1. Moreover, v is completely joined to all the ≥ − − vertices of H, giving:

ωM (G) ωM (H) + 1. ≥

Also, since S is universal in G, we have that ωM (H) ωM 1. ≤ − Finally, by applying the induction hypothesis for H we obtain:

m ωM (G) ωM (H) + 1 F (ωM (H))+1 ≥ ≥ 2 − ≥ n f(ωM ) 1 n − − F (ωM 1) + 1 F (ωM ), ≥ 2 − − ≥ 2 − completing thus the proof.

58 We also obtain the following:

Corollary 3.1.13 Any graph G satisﬁes:

n c 2 ωM (ωM ) log ωM , ≥ 2 − 2 p where c is the global constant from 3.1.6.

3.2 Clique Minors in Graph Complements

In this section we will provide a proof for Theorem 3.1.3, stated at the beginning of the previous section. We can actually show something stronger:

Theorem 3.2.1 There exists a function g : N∗ (0, ) such that for any → ∞ k 1 and any graph G with ωM (G),ωM (G) k + 1 and V (G) g(k) we ≥ ≥ | | ≥ have: k ωM (G)ωM (G) V (G) . ≥ 2| |

Proof. Let n = V (G) , s = ωM (G) and t = ωM (G). Without loss of | | generality we may assume that s t. From Lemma 3.1.11, we obtain that: ≥ a n s . ≥ 2 √log n

59 k k If t √log n, then st n, so the statement of the theorem holds. ≥ a ≥ 2 Therefore, we may assume that:

k t< log n. (3.1) a p By using Corollary 3.1.13 and t k + 1, we obtain: ≥ k + 1 k + 1 st n c t2 log t. (3.2) ≥ 2 − 2 p Let g(k) be the smallest integer such that:

ck2(k + 1) k 1 n 2 log n log + log log n, (3.3) ≥ a r a 2 for any n g(k). ≥ By combining (3.1), (3.2) and (3.3) we obtain:

kn st , ≥ 2 and the proof is complete.

We are ﬁnally in a position to prove Theorem 3.1.3, i.e. that any graph G large enough satisﬁes:

ωM (G)ωM (G) V (G) , ≥ | |

by applying the previous theorem.

Proof. Since Hadwiger’s Conjecture is trivially true for graphs without a K minor, we may assume that ωM (G) 3 and ωM (G) 3. By letting 3 ≥ ≥ k = 2 in Theorem 3.2.1, we obtain the desired inequality.

60 3.3 Complements of C4-free Graphs

Excluding an induced cycle of length four from the complement of the graph suﬃces to force the graph to satisfy Hadwiger’s Conjecture, as shown by the following:

Theorem 3.3.1 Let G be a graph such that G is C4-free. Then G satisﬁes Hadwiger’s Conjecture.

Proof. We proceed by induction on n = V (G) . We may assume that | | n 4 and that the theorem holds for graphs with fewer than n vertices. Since ≥

G contains no induced C4, it follows that G contains no induced matching of size two.

If G is P3-free, according to Theorem 1.2.5, G is perfect, so it obviously satisﬁes Hadwiger’s Conjecture. We may assume therefore that G contains an induced path of length three, say P = v0v1v2v3. Let e = v0v1 and f = v2v3 be the ﬁrst and the last edges of the path P .

Let A be the set of all the vertices not joined to e and B be the set of all the vertices not joined to f. Since G does not have an induced 2-matching,

A and B are independent sets. Let H be the induced subgraph obtained from G by deleting A, B and v0,v1,v2,v3. Any vertex of H is joined to both e and f, therefore:

ωM (G) 2+ ωM (H). ≥ We claim that the vertices in A, B and P can be properly 2-colored.

Indeed, color v0 and v2 red and v1 and v3 blue. Since G has no induced

2-matching, v3 has no neighbor in A and v0 has no neighbor in B.

61 Moreover, both A and B are stable sets, we can color all the vertices of

A blue and all the vertices of B A red. It follows that: \ χ(G) 2+ χ(H), ≤ so by using the induction hypothesis for H and the last two inequalities, we obtain χ(G) ωM (G), q.e.d. ≤

3.4 Excluding Clique Minors in the Comple-

ment

An interesting approach for gaining further insight into Hadwiger’s Conjec- ture would be to try to bound the clique-minor number of the complement of a graph. We have the following:

Theorem 3.4.1 Let G be a graph such that ωM (G) 3. Then G satisﬁes ≤ Hadwiger’s Conjecture.

Proof. We proceed by induction on n = V (G) . We may assume that n 4 | | ≥ and that the statement holds for graphs smaller than G.

Since G does not contain a K -minor, it follows that δ(G) 2, so G has 4 ≤ a vertex v such that NG(v) n 3. Let A := G ( v NG(v)) and H the | | ≥ − \ { }∪ subgraph of G induced by NG(v). We know that A 2. | | ≤ If A is an independent set, we have:

χ(G) 1+ χ(H) 1+ ωM (H) ωM (G), ≤ ≤ ≤ so the theorem holds for G.

62 Otherwise, we may assume that A = u , u and e = u u is an edge of { 1 2} 1 2 G. Let B be the set of all the vertices in H joined to the edge e, i.e. the set of all vertices with at least one neighbor in A, and let C := H B. \ If C is empty, the edge e is universal for H. If there exists a vertex u H ∈ such that u is adjacent to u and non-adjacent to u , v, u A is 2-colorable 1 2 { }∪ and we obtain:

χ(G) 2+ χ(H u) 2+ ωM (H u) ωM (G), ≤ \ ≤ \ ≤ since the edges uv and e are joined and universal to H u. If both u and \ 1 u2 are completely joined to H, by a similar argument we have:

χ(G) 2+ χ(H) 2+ ωM (H) ωM (G). ≤ ≤ ≤

Therefore, we may assume that C is non-empty. In this case, C is a clique, since if x, y C are non-adjacent, the vertices v, u , u , x, y produce ∈ 1 2 a subdivided K4 in the complement of G. Moreover, by the same argument, if C 2, any edge in C is universal for H. | | ≥ The case C = 1 needs to be treated separately. Let C = x and | | { } assume that there exists a vertex u H such that u is adjacent to u and ∈ 1 non-adjacent to u2. If u is not adjacent to x, then we can use uvx and e as nodes, since they are universal for the rest of the graph and the subgraph induced by all ﬁve of them is 2-colorable. If u and x are adjacent, we can use v and uu1u2 as nodes. If such a vertex u does not exist, by symmetry u1 and u are completely joined to H x. and we proceed like before. 2 \

63 Finally, assume that C 2. If all the vertices in C are universal for H, | | ≥ we have, since there are no edges between u , u and v C: 1 2 { }∪

χ(G)= χ(G u , u ) ωM (G). \ { 1 2} ≤

Otherwise, let z be a non-neighbor of some vertex x C. Let y C ∈ ∈ such that y = x. Then z is universal to C and we can use v, xy and zu u 6 1 2 as nodes, noting that the subgraph induced by all these vertices is always

3-colorable. The proof is now complete.

64 CHAPTER 4 GRAPHS WITH EDGES OF INTEGER LENGTH

The question of whether any ﬁnite graph can be embedded in the Eu- clidean 3-space such that all its edges are straight line segments of integer length was suggested by Dan Archdeacon in his list of Problems in Topolog- ical Graph Theory. We will devote this whole chapter to proving that such an embedding is indeed possible.

Since any graph G on n vertices is a subgraph of the complete graph Kn, it suﬃces to ﬁnd an embedding for Kn only. Also note that if all the edges are straight line segments of rational length, we can obtain an embedding with edges of integer length by using a suitable scaling transformation.

65 4.1 Preliminaries

We start with a few deﬁnitions and auxiliary results which will be used to prove the main theorem.

Lemma 4.1.1 For any M N, there exists β = β(M) Q (0, 3) such ∈ ∈ ∩ that the equation:

9x4 + 8βx3 + 34x2 8βx +9= y2 (4.1) − has at least M distinct rational solutions.

Proof. We study ﬁrst the general equation:

ax4 + bx3 + cx2 + dx + e = y2, (4.2) where all the coeﬃcients a, b, c, d and e are rational numbers.

From a particular solution (x , y ) Q Q, we can construct a new 0 0 ∈ × rational solution as follows: let x = x + X with X = 0 in (4.2), then we 0 6 obtain the new equation:

4 3 2 2 3 2 2 2 aX +X (4ax0 +b)+X (6ax0 +3bx0 +c)+X(4ax0 +3bx0 +2cx0 +d)+y0 = y . (4.3)

′ ′ ′ 2 ′ 3 2 Now let a = a, b = 4ax0 + b, c = 6ax0 + 3bx0 + c, d = 4ax0 + 3bx0 +

2cx0 + d, and equation (4.3) becomes:

′ 4 ′ 3 ′ 2 ′ 2 2 a X + b X + c X + d X + y0 = y . (4.4)

2 By letting y = AX + BX + y0, (4.4) becomes:

′ 4 ′ 3 ′ 2 ′ 2 2 4 3 2 2 2 a X +b X +c X +d X+y0 = A X +2ABX +X (B +2Ay0)+2By0X+y0. (4.5)

66 2 ′ ′ If A and B are chosen such that B + 2Ay0 = c and 2By0 = d , X must satisfy: a′X + b′ = A2X + 2AB,

so ﬁnally we get: b′ 2AB X = − , A2 a′ 2 − d c B where B = ′ and A = ′ . 2y0 2−y0 2 The new solution for the equation (4.2) is therefore (x0 +X, AX +BX +

2 ′ y0), with X, A and B defined as above. Since all the expressions y0, b , AB, A2 can be expressed as rational fractions in the indeterminates a, b, c, d and e, it follows that the new solution of the equation (4.2) can be expressed also D(a, b, c, d, e) as where D and E are polynomials in five indeterminates with E(a, b, c, d, e) rational coefficients.

We now return to equation (4.1). We start with the particular solution

(x0 = 0, y0 = 3) and we construct by the above algorithm new solutions

(x , y ), (x , y ), , (xM , yM ), where: 1 1 2 2 ···

Di(β) xi = Ei(β) with Di,Ei Q [X] for any 1 i M. It remains to be shown that we can ∈ ≤ ≤ chose β (0, 3) Q such that the generated solutions are all distinct (more ∈ ∩ precisely, the x’s are distinct).

67 In order to achieve this, we give the following definitions: D(X) For any rational fraction F (X) = with D(X) = a Xm + + E(X) 0 ··· k am Q [X], E(X)= b X + + bk Q [X] and a , b = 0, we define +1 ∈ 0 ··· +1 ∈ 0 0 6 a the degree of F by deg(F )= m k and the content of F by c(F )= 0 . − b0 The invariants above are well-defined and we have the following properties: (i) deg(F + G) =max(deg(F ), deg(G)) if deg(F ) =deg(G). 6 If deg(F ) =deg(G) and c(F )+c(G) = 0, the equality still holds and we 6 also have c(F + G) =c(F )+c(G); (ii) if deg(F )

(iii) deg(FG) =deg(F )+deg(G), c(FG) =c(F )c(G); F F c(F ) (iv) deg =deg(F ) deg(G), c = if G = 0. G − G c(G) 6 µ ¶ µ ¶

Thinking of the solutions xi as elements in the fraction ﬁeld Q(β), we will show that all xi are distinct. Using the construction presented above, xi+1 is constructed based on xi as follows:

xi+1 = xi + X b′ 2AB X = − , A2 a′ − where: d′ B = 4 3 2 2 9x 8βx + 34x + 8βxi + 9 i − i i c′ B2 A = p − , 4 3 2 2 9x 8βx + 34x + 8βxi + 9 i − i i ′ ′ ′ 2 ′ 3 2 and a = a = 9, b = 4axi 8β, c = 6ax 24β + 34, d = 4ax 24βx + p− i − i − i 68xi + 8β.

68 For i = 0 we obtain successively, using the properties (i)-(iv) above: deg(d′) = 1, deg(B) = 1, deg(c′) = 0, deg(A) = 2, deg(X) = 2, deg(x ) = − 1 2. Also we can compute c(x ) = 9. − 1 We will show by induction on i that deg(xi) = 2 and c(xi) = 9i. For − i = 1 the assertion is true. We can assume it true for a certain i and show that it holds for i + 1. We obtain succesively that deg(AB) = 2 and deg(A2) = 4, 2c(A)c(B) so deg(X) = 2 4=2 and c(X)=− 2 = 9. − c (A) Since c(xi)+ c(X) = 0, we get deg(xi ) = 2 and c(xi ) = c(xi)+ 6 +1 − +1 c(X)=9(i + 1), so the assertion holds for i + 1.

From the above discussion, we obtain that the elements xi viewed as elements of the fraction ﬁeld Q(β), are pairwise distinct. We consider the M equations in β given by: 2 µ ¶

xi = xj, (4.6) for any pair (i, j) with 1 i < j M. Each of the above equations is ≤ ≤ an equality of two non-identical rational fractions, so each has finitely many M solutions. The union of the solution sets of the equations is still a 2 µ ¶ finite set. Since Q (0, 3) is infinite, there exists a β that is not a solution ∩ for any of the equations (4.6). This element will give distinct solutions for the initial equation (4.1), q.e.d.

69 We deﬁne the following geometrical transformation in space: ﬁx a point

T in space; for any point U in space, deﬁne U ′ = φ(U) to be the unique point satisfying:

(i) U ′ lies on the segment T U and is on the same side of T as U;

(ii) T U T U ′ = 1. | | · | | This transformation will be called the inversion with respect to the point T and satisﬁes the following:

Lemma 4.1.2 Let U a point in space and a plane that does not contain U. P The image of through the inversion with respect to the point U is contained P in a sphere that passes through U.

Proof. By a series of convenient translations, rotations and scalings we may assume that the point T is the origin of the coordinates system and the plane has the equation x = 1. For a point U(1,y,z) , the image P ∈ P U ′ = φ(U) has coordinates (x′, y′, z′), where: 1 x′ = 1+ y2 + z2 y y′ = 1+ y2 + z2 z z′ = , 1+ y2 + z2 and it can easily be checked that we have:

(x′ 1/2)2 +(y′)2 +(z′)2 = 1/4, − and therefore the image U ′ lies on the sphere centered at (1/2, 0, 0) with radius 1/2.

70 4.2 The Main Result

We are now ready to state and prove the main result of this chapter:

Theorem 4.2.1 For any n 2, the complete graph Kn can be embedded in ≥ R3 such that the edges are straight line segments of integer length.

Proof. As noted at the beginning of the chapter, it suﬃces to embed the complete graph Kn such that all the edges have rational lengths.

Consider the points Pi(cos αi, sin αi, 0) for i N and Q(a,β,c). We will ∈ show that we can choose reals αi (0, 2π) for i N such that all the distances ∈ ∈ PiPj are rational, for any distinct positive integers i and j. Also we can | | choose a,β,c such that all the distances QPi are rational and distinct for | | all i N. ∈ For any i, j N we have: ∈

αi αj αi αj αi αj PiPj = 2 sin − = 2 sin cos cos sin , | | 2 2 2 − 2 2 ¯ ¯ ¯ ¯ ¯ αi ¯ αi¯ ¯ so it suﬃces to have¯ sin ,¯ cos ¯ Q for any i N. ¯ ¯ 2 ¯ 2 ∈ ∈

αi In order to achieve this, it suﬃces to have tan Q for any i N, since: 4 ∈ ∈ αi αi 2tan 4 sin = 2 αi 2 1+tan 4 and 2 αi αi 1 tan 4 cos = − 2 αi . 2 1+tan 4

71 αi Let ti = tan for i N. Then we may write: 4 ∈

2 2 2 QPi = (a cos αi) +(β sin αi) + c = − − 2p 2 2 = a + β + c + 1 2a cos αi 2β sin αi. − − p Since

2 4 2 2 αi 2ti ti 6ti + 1 cos αi = 1 2 sin = 1 2 = − − 2 − 1+ t2 (1 + t2)2 µ i ¶ i and 2 αi αi 2ti 1 ti sin αi = 2 sin cos = 2 2 − 2 , 2 2 1+ ti 1+ ti we obtain:

2 2 2 2 2 4 2 3 (a + β + c + 1)(1 + ti ) 2a(ti 6ti + 1) 8β(ti ti ) QPi = − 2 − − − . p 1+ ti By letting a = 1, c = 9 β2 with β (0, 3), the above simpliﬁes to: − ∈ p 4 3 2 9ti + 8βti + 34ti 8βti + 9 QPi = 2 − , p 1+ ti for any i N. ∈ It follows that in order to have QPi Q for some i N, it suﬃces to ∈ ∈ require that ti is a rational solution of the equation:

9x4 + 8βx3 + 34x2 8βx +9= y2, (4.7) − which is exactly the equation in the statement of Lemma 4.1.1.

72 We claim that the following statement holds:

For any M ′ N, there exists β = β(M ′) Q (0, 3) such that there exist ∈ ∈ ∩ distinct rationals t ,t , ,tM ′ with the property that the corresponding 1 2 ··· points P , P , , PM ′ satisfy: 1 2 ··· (i) the distances QPi are all rational; | | ′ (ii) QP = QPj for any 1 i < j M . | 1| 6 | | ≤ ≤ M ′ Indeed, take M in Lemma 4.1.1 such that M > k=1 k, so there exists β Q (0, 3) such that equation (4.1) has distinctP rational solutions ∈ ∩ s ,s , ,sM . 1 2 ··· Choose t = s . Denote by the circle with center (0, 0, 0), radius 1 and 1 1 C lying in the plane XOY . In order to choose t such that QP = QP , the 2 | 2| 6 | 1| point P corresponding to t should avoid the intersection of and the sphere 2 2 C with center Q and radius QP . Because (a, β) = (0, 0), this intersection con- 1 6 tains at most two points, one of them being P1. From the M solutions of the equation (4.1), eliminate the eventual solution corresponding to the second point in the intersection, then choose t2 to be one of the remaining solutions.

′ Suppose now that we have chosen t ,t , ,tq with q M 1 and we 1 2 ··· ≤ − q want to choose tq . At this stage, we have at least M k remaining +1 − k=1 solutions of equation (4.1). In order to ensure that QPq P= QPi , for all | +1| 6 | | 1 i q, the point Pq must avoid a set consisting of at most q points, so ≤ ≤ +1 tq+1 must avoid a set with at most q elements. From the condition imposed on M, we can indeed choose tq+1.

73 We proved the following:

For any M N, there exist t ,t , ,tM such that the corresponding ∈ 1 2 ··· points P , P , , PM have the following properties: 1 2 ···

Let φ be the inversion with respect to the point Q and let Ri = φ(Pi) for

1 i M. We will show that RiRj Q for any 1 i < j M. ≤ ≤ | |∈ ≤ ≤

We have, for any i = j: 6

PiPj QPi QPi | | = | | = | | = QPi QPj , RiRj QRj 1 | | · | | | | | | QPj | | and therefore:

PiPj RiRj = | | Q. | | QPi QPj ∈ | | · | |

In order to ﬁnish the proof, we will show that the above constant M can be chosen conveniently (depending of n) such that among the points

R ,R , ,RM , we can ﬁnd n distinct points S ,S , ,Sn such that no 1 2 ··· 1 2 ··· four of them are in the same plane.

74 Figure 4.1: Embedding Kn into 3-Space.

This way, we can embed the graph Kn with the requested properties by placing each vertex in one of the points S ,S , ,Sn. 1 2 ···

n Choose M such that M > n . k=1 3 µ ¶ Since all the points Pi lie inP the plane XOY , by using Lemma 4.1.2, we deduce that all the points Ri lie on a sphere, say . Choose S = R ,S = R S 1 1 2 2 and S3 = R3.

In order to choose S4 such that S1,S2,S3 and S4 are not in the same plane, denote by the plane that passes through the non-collinear points S ,S ,S A 1 2 3

75 and also denote by the cone having the generator family QP : P , i.e. T { ∈ C} the cone with vertex Q and generators passing through the circle containing all the points Pi. Obviously all the points Ri lie on . T The intersection of , and contains at most 4 points, three of them A T S being S ,S ,S . Indeed, plane is not parallel with the plane of the circle 1 2 3 A containing the points Pi since all the distances QPi are distinct, so the intersection of and is an ellipse which is not a circle. But and have A T E E S in this case at most four points in common. Eliminate the point Rj that lies in the intersection (in the case that such a j exists) and chose S4 from the remaining points.

Suppose now that we have chosen the points S ,S , ,Sq and we want 1 2 ··· to choose Sq+1. A similar argument as above shows that Sq+1 must avoid q at most of the remaining points R (for each triple (S ,S ,S ) with 3 i i j k µ ¶ 1 i

76 CHAPTER 5 CONCLUSION AND FURTHER RESEARCH

Some of the oldest problems in Graph Theory involve colorings of graphs, with the Four Color Theorem being the classical example. Motivated in part by this, Hadwiger made his famous conjecture that any graph without a

Kk+1-minor is k-colorable. Even for k = 4 and k = 5 the proof of the conjecture is extremely diﬃcult and uses computers in an essential way. Another way of attempting to make further progress in solving the conjecture in general is to bound other invariants of the graph.

The most likely candidate for such an approach is the stability number of a graph. The problem becomes very diﬃcult even for the α = 2 case, since we have to be able to show the existence of a clique minor of size at least half the number of vertices of the graph.

77 As shown earlier, in order to achieve this, we have to be able to build a universal matching covering at least one third of the vertices of the graph. Without imposing additional conditions, such as excluding certain graphs as induced subgraphs or asking that the edge density is high, ﬁnding such a matching proves to be extremely diﬃcult. The main problem arises from the lack of a suitable structure theory for graphs with α = 2, even though they (and their complements, the triangle-free graphs) have been studied extensively in the past.

The inequality involving the clique minor number of a graph and its complement, proved in Chapter 3, is a direct consequence of Hadwiger’s

Conjecture. In general, bounding the clique minor number in the complement of a graph forces the graph itself to have a clique minor of size almost half the number of vertices, and these graphs should easily satisfy the conjecture. To gain further insight into the general case, it would be interesting to see what happens for graphs with α = 3 say, if we exclude C4 or C5 as induced subgraphs. Even if we were able to show that all such graphs contain a clique minor of size at least a third of the number of vertices, the conjecture would still not be settled for this special case. In the α = 2 situation, there exists a result proving the equivalence of these statements, a result that is not known for higher values of the stability number.

In some instances, for example when excluding K4 as a minor in the complement, we were able to prove the conjecture directly, without showing the existence of a large clique minor ﬁrst. For higher cases, such an approach

78 becomes increasingly diﬃcult. The path towards the solution for the general case of Hadwiger’s Conjecture probably involves all these possible techniques. The α = 2 case is still open, and it is very likely that signiﬁcant progress here will yield invaluable insight for the general case. Even showing that there exists a linear bound for the chromatic number in terms of the clique minor number is open, and it is not known whether such a bound would bring us any closer to the solution for Hadwiger’s Conjecture.

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