GRAPH MINORS AND HADWIGER S CONJECTURE

Transcription

1 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 Professor Dijen Ray-Chaudhuri Adviser Graduate Program in Mathematics

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3 ABSTRACT One of the central open problems in Graph Theory is Hadwiger s Conjecture, which states that any graph with no K k+1 -minor is k-colorable. Restated, 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 C 4 or an induced C 5, 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

4 In the memory of my mother iii

5 ACKNOWLEDGMENTS Any writing is the product not only of its authors, but also of the environment where the authors work, of the encouragements and critics gathered from colleagues and teachers and conversations after seminars and conferences. 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 developed 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

6 VITA January 29, Born - Constanta, Romania B.S. Mathematics, University of Bucharest M.S. Mathematics, University of Bucharest present... Graduate Teaching Assistant, The Ohio State University FIELDS OF STUDY Major Field: Mathematics v

7 TABLE OF CONTENTS ABSTRACT... ii DEDICATION... iii ACKNOWLEDGMENTS... VITA... iv v LIST OF FIGURES... viii 1 INTRODUCTION Basic Concepts Graph Colorings Graph Minors and Hadwiger s Conjecture Large Clique Minors and Connections with Other Invariants Upper Bounds for the Chromatic Number vi

8 2 HADWIGER S CONJECTURE FOR GRAPHS WITH STA- BILITY NUMBER Preliminaries Lower Bounds for the α = 2 Case Graphs with Low Connectivity Excluding Certain Subgraphs Graphs with High Edge Density GRAPH COMPLEMENTS Preliminaries Clique Minors in Graph Complements Complements of C 4 -free Graphs Excluding Clique Minors in the Complement GRAPHS WITH EDGES OF INTEGER LENGTH Preliminaries The Main Result CONCLUSION AND FURTHER RESEARCH BIBLIOGRAPHY vii

9 LIST OF FIGURES 1.1 Constructing a Universal Subgraph The C 4 -free Case The values of f and F for n Embedding K n into 3-Space viii

10 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 1,S 2,,S k } of disjoint subsets of S is called a partition of S if S = k i=1 S i. The sets S 1,S 2,,S k are called the classes of the partition. We denote by P k (S) the set of all the subsets of S of cardinality k. 1

11 Definition A graph is a triple G = (V,E,I), where V and E are disjoint sets and I is a mapping I : E P 1 (E) P 2 (E). 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) P 1 (E); two distinct edges e,e E are called parallel if I(e) = I(e ). 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 P 2 (V ). Unless otherwise stated, all the graphs considered in 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 P 2 (V ). In this case, we write 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

12 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 An independent (or stable) set is a set of pairwise nonadjacent 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 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 N G (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 N G (S). The degree d G (v) of a vertex v V (G) is the number of edges incident with v; for simple graphs, it is also the cardinality of N G (v). The minimum degree of G, denoted by δ(g), is defined to be δ(g) := min {d G (v) : v V (G)}. Similarly, the number (G) := max {d G (v) : v V (G)} is called the maximum degree of the graph G. 3

13 Definition The average degree of a graph G is given by: d(g) := 1 V (G) v V (G) Remark Clearly δ(g) d(g) (G). d G (v). Definition A path is a graph P = (V,E) of the form: V = {v 0,v 1,,v k } and E = {v 0 v 1,v 1 v 2,,v k 1 v k }, where the v i s are all distinct. The path P defined above is denoted by P = v 0 v 1 v k. The vertices v 0 and v k 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 P k. If P = v 0 v 1 v k 1 is a path with k 3, then the graph C = P v k 1 v 0 is called a cycle and we also write C = v 0 v 1 v k 1. The length of the cycle is the number of its edges; a cycle of length k is denoted by C k. 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 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

14 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 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 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

15 Definition 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 (König) 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 p 1 (G) be the number of odd components of G. The existence of a perfect matching in general graphs is described by the following: Theorem (Tutte) A graph G has a perfect matching if and only if p 1 (G \ S) S for all S V (G). 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

16 The following theorem, whose proof can be found in Section 2.2 of [3], describes a matching structure on an arbitrary graph: Theorem 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 Turán s Theorem. Definition 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 = K r for some r 2, all the complete (r 1)-partite graphs are edge-maximal without containing K r. 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 Turán graphs and are denoted by T r 1 (n). Also we denote by t r 1 (n) the number of edges of T r 1 (n). 7

17 Theorem (Turán) For all integers r,n with r 2, every graph G not containing K r, with n vertices and ex(n,k r ) edges is a T r 1 (n). Proof. See, for instance, Section 7.1 in [3]. The following upper bound will be useful later: Proposition For any integers r,n with r 2 we have: t r 1 (n) 1 2 n2r 2 r 1. Proof. Let A 1,A 2,,A r 1 be the classes of an (r 1)-partition of T r 1 (n). We have: ( ) n r 1 ( ) Ai t r 1 (n) = n r 1 A i 2 2 i=1 i=1 ( n2 2 1 r 1 ) i=1 (r 1) A 2 i = 1 2 r 1 2 n2r 2 r 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 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. 8

18 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 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 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

19 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 A graph G is called P 3 -free if it does not contain any induced path of length 3. The class of P 3 -free graphs provides a well-known subclass of graphs which are perfect: Theorem All P 3 -free graphs are perfect. Proof. Let G be a P 3 -free graph. We proceed by induction on n = V (G). 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 1,C 2,,C k with k 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 C 1. Since G is connected, it follows that there exists and edge e = xy of G with both endpoints in C 1 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 C 2, say u. The vertices x,y,v,u, in this order, induce a path of length 3, contrary to the hypothesis. 10

20 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 For any graph G we have: χ(g) V (G) α(g). Proof. Let k = χ(g) and let C 1,C 2,,C k be the color classes induced 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 α(g) V (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

21 Graphs with few edges have low chromatic number, as shown by the following: Proposition Let e = E(G) be the cardinality of the edge set of the graph G. Then: χ(g) e. Proof. Let k = χ(g) and let C 1,C 2,,C k be the color classes induced by a k-coloring of G. For any two different color classes C i and C j, 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 Solving this inequality in terms of k, we obtain the conclusion. Definition 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 admits a k-edge-coloring, we say that G is k-edge-colorable. Definition The chromatic index of a graph G, denoted by χ e (G), is defined by: χ e (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

22 Theorem (Vizing) If G is a simple graph, then: (G) χ e (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 A class G of graphs is called closed under disjoint union if for any two graphs G and H from G, not necessarily distinct, their disjoint union is also a member of G. Definition Let k be a positive integer. A class G of graphs is called k-colorable if any graph G in G is k-colorable. Definition 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 Let k be a positive integer and f : N (0, ) be a function. A class G of graphs is called (k,f)-colorable if all its members are (k, f)-colorable. Theorem Let k be a positive integer and f : N (0, ) be a function such that lim f(n)/n = 0 as n. Let G be a graph class that is closed under disjoint union and is (k,f)-colorable. Then G is k-colorable. 13

23 Proof. Let G be a member of G with n = V (G) and let p be a positive 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 G and H is (k,f)-colorable. Since V (H) = np, all but at most f(np) 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 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 Conjecture 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

24 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 N G (u) N G (v). Definition 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 1,v 2,,v k } is a minor of G if there exist connected vertexdisjoint subgraphs of G, say C 1,C 2,,C k, with the following property: if v i and v j are adjacent in H for some 1 i < j k, then there exists an edge e E(G) joining C i and C j. The subgraphs C 1,C 2,,C k are called the nodes of the minor. 15

25 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 1,C 2,,C k such that any two of them are joined by an edge. By analogy to the clique number of a graph, we provide the following: Definition Let G be a graph. The clique-minor number of G, denoted by ω M (G), is defined by: ω M (G) = max{k : K k is a minor of G}. One of the central open problems in graph coloring is Hadwiger s Conjecture, which relates the chromatic number to the size of the largest clique minor contained in a graph: Conjecture (Hadwiger) Any graph G which does not contain K k+1 as a minor is k-colorable. Using the above notation, the conjecture can be restated as: Conjecture For any graph G we have: χ(g) ω M (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 K 2 minor is edgeless, and therefore 1- colorable. If k = 2, graphs with no K 3 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

26 For k = 3, graphs with no K 4 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

27 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 For a graph G with e = E(G) we have: ω M (G) e. Proof. Let k = ω M (G). According to a previous remark, we can find connected, vertex-disjoint subgraphs of G, C 1,C 2,,C k say, such that any two of them are joined by an edge. It follows that G has at least k(k 1) 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 K 5, 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 For any graph G we have: ω M (G) V (G) + ω(g). 2 Proof. Let n = V (G), k = ω M (G) and C 1,C 2,,C k be vertex-disjoint connected subgraphs of G, pairwise joined by an edge. Without loss of generality, we may assume that C 1 = C 2 = = C p = 1 and C i 2 for 18

28 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 k C i 2(k p) + p = 2k p, i=1 and therefore: ω M (G) n + p. 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 ω M (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 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 For any graph G we have: ω M (G) V (G) α(g). We will prove the following weakening of the above conjecture: Theorem Any graph G satisfies: ω M (G) V (G) 2α(G) 1. 19

29 We need a few definitions and auxiliary results. Definition 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 If H is a universal subgraph of a graph G, we have: ω M (G) ω M (G \ H) + 1. Proof. Let k = ω M (G \ H) and N 1,N 2,,N k be the nodes of a clique 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 1,N 2,,N k. Therefore, we can obtain a clique minor of size k + 1 in G with nodes H,N 1,N 2,,N k. The following lemma shows another way of obtaining clique minors: Lemma 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: ω M (G) ω M (H) + ω M (K). 20

30 Proof. Let k = ω M (H), p = ω M (K) and M 1,M 2,,M k ; N 1,N 2,,N p 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 1,M 2,,M k and N 1,N 2,,N p. 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 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\(N G (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 1,C 2,,C k, with k 1. 21

31 Since G is connected, there exist edges e 1,e 2,,e k E(G) such that e i has an endpoint in C i, say v i, and one endpoint in N G (v), say u i, for all 1 i k. The vertices u i (1 i k) are not necessarily distinct. Applying the induction hypothesis to the connected graphs C 1,C 2,,C k and vertices v 1,v 2,,v k, we deduce the existence of universal subgraphs H 1,H 2,,H k for each of the graphs C 1,C 2,,C k. Moreover, we have v i V (H i ) and V (H i ) 2α(H i ) 1 for all 1 i k. Figure 1.1: Constructing a Universal Subgraph. 22

32 Let H be the subgraph of G induced by {v} k i=1 {u i} k i=1 V (H i). Then it is easy to check that H is universal for G and: V (H) 1 + k + since: k V (H i ) 1 + k + i=1 k (2α(H i ) 1) 2α(H) 1, i=1 thus providing the conclusion. α(h) 1 + α(h 1 ) + α(h 2 ) + + α(h k ), Corollary Any connected graph G has a universal subgraph with at most 2α(G) 1 vertices. We also need the following general result: Lemma Let k be a positive integer and a 1,a 2,,a k and b 1,b 2,,b k be strictly positive reals. Then: a i max a 1 + a a k. 1 i k b i b 1 + b b k Proof. Without loss of generality, we may assume that a 1 a i = max, b 1 1 i k b i i.e. a 1 b i a i b 1 for all 1 i k. By adding these k inequalities side by side, we obtain: a 1 (b 1 + b b k ) b 1 (a 1 + a a k ), which is equivalent to the statement of the lemma. 23

33 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 1,C 2,,C k. Let α = α(g) and n i = V (C i ) and α i = α(c i ) for 1 i k. Then α 1 + α α k = α. The largest clique minor in G is completely contained in one of the components C 1,C 2,,C k. Using the induction hypothesis and Lemma , we obtain: ω M (G) max 1 i k yielding thus the same conclusion for G. n i 2α i 1 n 2α k, 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 and the induction hypothesis for F, we obtain: ω M (G) ω M (F) + 1 V (G) 2α(G) + 1 2α(F) V (G) 2α(G) 1, showing that the graph G also has the desired property. 24

34 1.5 Upper Bounds for the Chromatic Number The following result provides a lower bound for the clique minor number in terms of the average degree of a graph: Theorem (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 K k minor. Up to the value of c, this bound is best possible as a function of k. The following is an immediate consequence: Corollary There exists a constant c > 0 such that any graph G satisfies: χ(g) c ω M (G) log ω M (G). 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 Any graph G admits an induced subgraph H such that V (G) χ(h) ω M (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

35 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 2 and with χ(h ) ω M (G \ U). 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 + ω M (G \ U) ω M (G), since U is universal. On the other hand: V (H) A + V (H ) V (U) 2 + V (G \ U) 2 = V (G), 2 which shows that the statement holds for G. Theorem For any graph G with n = V (G) and ω M = ω M (G) we have: χ(g) ω M ( log 2 n ω M ) + 1. Proof. From the previous lemma, we have the existence of a ω M - colorable induced subgraph of G, say H 1, such that: V (G \ H 1 ) n 2. Let G 2 = G \ H 1, by applying the lemma again for G 2, we obtain a new induced subgraph H 2 with χ(h 2 ) ω M (G 2 ) ω M and such that G 2 \ H 2 has at most n 4 vertices. 26

36 Continuing this process, after p steps we have constructed vertex-disjoint subgraphs H 1,H 2,,H p, each of them ω M -colorable, such that: V (G \ This implies that: p H i ) n 2 p. i=1 If we choose p = conclusion. log 2 χ(g) p ω M + n 2 p. n, it follows that n 2 ω p M and we obtain the ω M 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 For any graph G with n = V (G), ω M = ω M (G) and α = α(g) we have: χ(g) ω M ( log 2 α + 2). n Proof. From Theorem 1.4.4, we obtain that 2α. This inequality, ω M together with the previous theorem, imply the desired result. 27

37 CHAPTER 2 HADWIGER S CONJECTURE FOR GRAPHS WITH STABILITY NUMBER Preliminaries It was stated earlier that Hadwiger s Conjecture is known for k 5, i.e. for graphs which do not contain K 6 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

38 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 For any graph G with α(g) = 2 we have: ω M (G) n 2, where n = V (G). It can be shown that for the α = 2 case, the above conjecture and Hadwiger 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 counterexamples 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 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 A class G of graphs is called closed under induced subgraphs if for any G G, any induced subgraph of G is also a member of G. 29

39 Theorem Let G be a graph class closed under induced subgraphs. Assume that ω M (G) for any G G. Then any graph of G satisfies V (G) 2 Hadwiger s Conjecture. V (G) Proof. Let G G such that ω M (G). Assume that the theorem 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 , there exists a set S such that G \S has odd components C 1,C 2,,C p with S < p. Also we may assume that G \ S has even components D 1,D 2,,D k, that S is matchable to G \ S and that C 1,C 2,,C p are critical. Let M = {u 1 v 1,u 2 v 2,,u m v m } be a matching of S into C 1,C 2,,C p, where u 1,u 2,,u m S and v i C i for any 1 i m (m < p). Also each of the graphs C i \ v i has a perfect matching, for any 1 i m. Let H be the subgraph of G induced by the vertices in G \ S. The subgraphs C 1,C 2,,C p and D 1,D 2,,D k are completely joined in G, so we have: k p ω M (G) ω M (C i ) + ω M (D i ). i=1 i=1 In any coloring of G \ S, the vertices of two different components of G \ S receive different colors. We can color each C i with V (C i) + 1 colors such that one color, say c i, 2 appears only at v i, for all 1 i p. Then we can use colors c 1,c 2,,c m 30

40 to color vertices u 1,u 2,,u m, so it follows that χ(g) = χ(g \ S). Since: χ(g \ S) k χ(c i ) + i=1 p χ(d i ) i=1 k ω M (C i ) + i=1 p ω M (D i ) ω M (G), i=1 we obtain the desired conclusion. According to classical Ramsey Theory, any graph satisfying α(g) = 2 contains a clique of size at least O( n log n) and examples have been constructed with clique numbers of this order of magnitude. Proposition 1.4.2, together with Hadwiger s Conjecture would imply that there exist graphs having clique minor numbers of size n 2 +O( n log n), essentially of the same order of magnitude as the lower bound of Conjecture 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 N G (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

41 Lemma 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 seagullfree. The following two remarks are obvious: Remark If α(g) = 2 and v V (G), then V (G) \ (N(v) {v}) is a clique. Remark 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 Let G be a graph with α(g) = 2. Then: ω M (G) V (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

42 If G contains a seagull H, then by lemma we have: ω M (G) ω M (G \ H) + 1 n = n 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 ω(g) n 2. With a little bit of additional work, it is possible to prove the following strengthening of the previous proposition: Proposition Let G be a graph with α(g) = 2. Then: ω M (G) V (G) + ω(g). 3 Proof. Let n = V (G) and K be a clique of G of size ω(g). Let S 1,S 2,,S k be vertex-disjoint seagulls of G, none of which having any ( vertex in K and such that H := G \ K ) k i=1 S i is seagull-free. In other words, we successively remove seagulls in G\K 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 k n 2ω(G) 3. Since any seagull is universal, by using the vertices of K and S 1,S 2,,S k as nodes, we obtain that: ω M (G) n 2ω(G) 3 + ω(g) = n + ω(g). 3 Otherwise, H is a disjoint union of two cliques K 1 and K 2, say. Any vertex in K is completely joined to at least one of the cliques K 1 and K 2, otherwise G 33

43 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 are graphs with α = 2 having the largest clique of size O( n log n). The n 3 is still essentially the best lower bound known for the general α = 2 case. In order to obtain an improvement of order O(n), we will impose additional density conditions in Section 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 Let G be a graph with α(g) = 2. Let k = k(g) be the connectivity of G. Then: ( ω M (G) min n k, n ), 2 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

44 α(g) = 2, G \C is a disjoint union of two cliques, say K 1 and K 2. Moreover, any vertex of C is completely joined with at least one of the cliques K 1 and K 2. Let C 1 be the set of all the vertices in C completely joined to K 1 and let C 2 := C \ C 1. Any vertex of C 2 completely joined to the clique K 2. Let m 1 = min(c 1,K 2 ) and m 2 = min(c 2,K 1 ). Define the bipartite graph B 1 = (V 1,V 2 ) as follows: V 1 = C 1, V 2 = K 2 and E(B 1 ) consists of all the edges of G with one endpoint in C 1 and the other in K 2. The bipartite graph B 2 is defined in a similar fashion on (K 1,C 2 ). We claim that B 1 contains a matching of size m 1. Otherwise, by König s Theorem, there exists a cover S for the edges of B 1 such that S < m 1 C 1. It is easy to check that S C 2 is a cut-set for G and S C 2 < C, giving a contradiction to the choice of C. The bipartite graph B 2 also contains a matching of size m 2. Let M 1 and M 2 be the matchings obtained above and let M = M 1 M 2 be their union. By symmetry, we only have to consider the following cases: Case 1: C 1 K 2, C 2 K 1. We can construct a clique minor by using the edges of M as nodes, so: ω M (G) m 1 + m 2 = K 1 + K 2 = n k. Case 2: C 1 < K 2, C 2 K 1. Then m 2 = K 1 and by considering the edges of M 2 and the vertices of K 2 as nodes, we get: ω M (G) m 2 + K 2 = n k. 35

45 Case 3: C 1 < K 2, C 2 < K 1. Without loss of generality we may assume that K 1 C 2 K 2 C 1. We can take the edges of M plus the vertices of K 1 \ C 2 as the nodes of the minor; thus: ω M (G) M + K 1 C 2 = C 1 + C 2 + K 1 C 2 = K 1 + C 1 n 2, and the proof is complete. Corollary If G is a graph with α(g) = 2 and connectivity at most V (G), then: 2 V (G) ω M (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 trianglefree 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 C 4 or C 5 free graphs (i.e. graphs not containing a C 4 or a C 5 as an induced subgraph) with α = 2 36

46 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 C 5 itself easily shows. Excluding C 5 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 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 1 and K 2 with no edges in between. Moreover, v is completely joined to at least one of the cliques K 1 and K 2. We may assume that v is complete to K 1. Since G is connected, there exists a vertex u V (K 2 ) N(v). It is easy now to check that the edge e = uv is universal in G. We can now show that a C 5 -free graph with α = 2 has a universal edge meeting a fixed maximal clique: Lemma 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

47 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 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 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 1 K 2, with K 1 and K 2 disjoint cliques. We may assume that b is completely joined to K 1 ; then v K 2. Choose any u K \ v, then u K 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

48 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 1 K 2 and v is completely joined to K 1, 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 1 and if v is completely joined to K 2, any u K 1 gives a universal edge e = uv for G. So we may assume that there exists a vertex v in K 2 such that v and v are not adjacent. If b is completely joined to K 2, choose any u K 1 and the edge ub is universal in G. Otherwise, there exists a vertex b K 2 such that b and b are not joined. It follows that b v, v is joined to b and b is joined to v. For any u K 1, the vertices u,v,b,v,b form an induced 5-cycle passing 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 C 5 -free graphs: Theorem Let G be a C 5 -free graph with α(g) = 2. Then: ω M (G) n 2, where n = V (G). Proof. Induction on n. Obvious for n = 1, 2, 3. If G is not connected, then G = K 1 K 2, with K 1,K 2 cliques, so the statement holds. Therefore 39

49 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: ω M (G) ω M (H) + 1 n = n 2. Corollary If G is a C 5 -free graph with α(g) = 2, then Hadwiger s Conjecture holds for G. Proof. This follows directly from Theorem via the standard reduction argument from Theorem We now turn our attention to the C 4 -free case. Graphs with α = 2, with no induced C 4, but containing an induced C 5 have a nice structure, as shown in the proof of the following: Theorem Let G be a C 4 -free graph with α(g) = 2. Then: ω M (G) n 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 = v 1 v 2 v 3 v 4 v 5 be such a cycle. 40

50 Since α(g) = 2, any vertex v G \ C is adjacent to either 3, 4 or 5 consecutive vertices of C. Let A i be the set of all vertices in G \ C joined in C only to v i,v i+1,v i+2 (including v i+1 itself) for i {1, 2,, 5}, where v 6 = v 1, v 7 = v 2. Figure 2.1: The C 4 -free Case. Let K i = A i A i+1 for i {1, 2,, 5}, where A 6 = A 1. It is easy to see that each K i is a clique in G. Let H be the subgraph of G induced by 5 i=1 A i and let m = V (H). We will show that ω M (H) m 2. 41

51 Let k = A 1 and assume that A i k, for any i {2, 3, 4, 5}. Then: K 2 + K 4 = m k, so we may assume that K 4 m k. We can obtain a clique minor of size 2 m 2 for H by using as nodes in the minor the vertices of K 4 and k disjoint induced subgraphs G 1,G 2,,G k, each with 3 vertices, one in A 1, one in A 2 and one in A 3. 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 C 4 -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 1. If u and v are not adjacent, then v 1,v,v 3 and u form an induced C 4 in G, contradiction. Denote by K the subgraph of G induced by B. It follows that K is completely joined to H. Then ω M (K) n m by the induction hypothesis, and by Lemma we obtain: ω M (G) ω M (H) + ω M (K) m 2 + n m 2 = n 2, so the statement of the theorem holds for G. 42