# Factors and Factorizations of Graphs

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1 Factors and Factorizations of Graphs Jin Akiyama and Mikio Kano June 2007 Version 1.0

2 A list of some notation G denotes a graph. X Y X is a subset of Y. X Y X is a proper subset of Y. X Y X \ Y, where Y X. X a X {a}, where a X. X + Y X Y, where X Y =. X Y (X Y ) (X Y ). X The number of elements in X. V (G) The set of vertices of G. E(G) The set of edges of G. G The order of G = V (G). G The size of G = E(G). deg G (v) The degree of a vertex v in G. N G (v) The neighborhood of v. N G [v] N G (v) {v}. N G (S) x S N G(x). E G (X, Y ) The set of edges of G joining X to Y. e G (X, Y ) The number of edges of G joining X to Y. Δ(G) The maximum degree of G. δ(g) The minimum degree of G. ω(g) The number of components of G. odd(g) The number of odd components of G. Odd(G) The set of odd components of G. iso(g) The number of isolated vertices of G. Iso(G) The set of isolated vertices of G. κ(g) The connectivity of G. λ(g) The edge connectivity of G. α(g) The independence number of G. α (G) The edge independence number of G. bind(g) The binding number of G. tough(g) The toughness of G. K n = K(n) The complete graph of order n. K n,m = K(n, m) The complete bipartite graph of order n + m. P n The path of order n. N The set of natural numbers = {1, 2, 3,...}. Z The set of integers. Z + The set of non-negative integers = {0, 1, 2,...}.

3 A list of some functions S and T denote disjoint vertex subsets of a graph G. δ(s, T ) = f(x)+ (deg G (x) f(x)) e G (S, T ) q(s, T ) x S x T = f(x)+ (deg G S (x) f(x)) q(s, T ), x S x T where q(s, T ) denotes the number of components C of G (S T ) such that x V (C) f(x)+e G(C, T) 1 (mod 2). δ (S, T ) = f(x)+ (deg G (x) f(x)) e G (S, T ) x S x T = f(x)+ (deg G S (x) f(x)) x S x T γ(s, T ) = f(x)+ (deg G (x) g(x)) e G (S, T ) q (S, T ) x S x T = f(x)+ (deg G S (x) g(x)) q (S, T ), x S x T where q (S, T ) denotes the number of components C of G (S T ) such that g(x) =f(x) for all x V (C) and x V (C) f(x)+e G(C, T) 1 (mod 2). γ (S, T ) = f(x)+ (deg G (x) g(x)) e G (S, T ) x S x T = f(x)+ (deg G S (x) g(x)) x S x T x V (C) f(x)+e G(C, T) 1 (mod 2). η(s, T ) = f(x)+ (deg G (x) g(x)) e G (S, T ) q(s, T ), x S x T where q(s, T ) denotes the number of components C of G (S T ) such that x V (C) f(x)+e G(C, T) 1 (mod 2).

4 Contents 1 Matchings and 1-Factors Matchings in Bipartite Graphs Covers and Transversals Augmenting Paths and Algorithms Factor Theorems Graphs Having 1-Factors Structure Theorem Algorithms for Maximum Matchings Regular Factors and f-factors The f-factor Theorem Regular Factors in Regular Graphs Regular Factors and f-factors in Graphs Regular Factors and f-factors in Bipartite Graphs (g, f)-factors and [a, b]-factors The (g, f)-factor Theorem Graphs Having the Odd-Cycle Property [a, b]-factors and (g, f)-factors [a, b]-factorizations Factorizations of Special Graphs Semi-Regular Factorization [a, b]-factorizations of Graphs Parity Factors Parity (g, f)-factors and (1,f)-Odd Factors (1,f)-Odd Subgraphs and Structure Theorem Partial parity (g, f)-factors and coverings H-Factors iii

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6 Preface Among the results in graph theory in the 18th century are Petersen s results on graph factors and factorizations. Petersen s Theorem states: Every even regular graph is 2-factorable. He proved it to approach a problem on Diophantine equations. Then he proved a second theorem. Every 2- connected 3-regular graph has a 1-factor. This arose from a counterexample he constructed to Tait s theorem : Every 3-regular graph which has no leaf is 1-factorable. This counterexample is the well-known Petersen graph. Petersen s theorems are byproducts of attempts to address other things. Perhaps Petersen himself had no inkling that these theorems would open the door to a very promising area of graph theory. Later came Hall s Marriage Theorem, a result obtained when he was studying the structure of subsets. Let G be a bipartite graph with bipartition (A, B). Then G has a matching that saturates A if and only if N G (S) S for all S A. König s Theorem followed: Every regular bipartite graph is 1-factorable. And then Tutte s 1-Factor Theorem: A graph G has a 1-factor if and only if odd(g S) S for all S V (G). These five theorems form the foundation of the study of factors and factorizations. König s theorem was the result of a conscious effort to answer a graph theory problem: Does every bipartite regular graph have a 1-factor? It seems that, at that time, König already had a good idea of how graph factors could be applied. In his paper On graphs and their applications in determinant theory and set theory, he starts by saying The following article deals with problems from analysis situs, the theory of determinants and set theory. He then continues to say that in fact, the notion of a graph and its usefulness as a method of representation actually relates these three disparate areas. A second stage of development is marked by Tutte s f-factor Theorem: Let G be a graph and f : V (G) Z + = {0, 1, 2, 3,...}. Then G has an f-factor if and only if for all disjoint subsets S and T of V (G), f(x)+ (deg G S (x) f(x)) q(s, T ) 0, x T x S v

7 vi where q(s, T ) denotes the number of components C of G (S T ) such that x V (C) f(x)+e G(C, T) 1 (mod 2); and Lovász s (g, f)-factor Theorem: Let G be a graph and g, f : V (G) Z such that g(x) f(x) for all x V (G). Then G has a (g, f)-factor if and only if for all disjoint subsets S and T of V (G), f(x)+ (deg G S (x) g(x)) q (S, T ) 0, x T x S where q (S, T ) denotes the number of components C of G (S T ) such that g(x) =f(x) for all x V (C) and x V (C) f(x) +e G(C, T) 1 (mod 2). Many subsequent theorems were proved based on these results. A third stage of development began in the 1980 s and is marked by the introduction of the notions of semi-regular factors and component factors. Since that time, many substantial results have been obtained and much progress has been made in this area. The results are detailed in this book. As far as we know, there is no comprehensive book on factors and factorizations. This is one compelling reason for writing this book. Since we wrote our survey paper entitled Factors and Factorizations of Graphs published in Journal of Graph Theory, vol. 9 (1985), we collected and analyzed almost all results in the area. In fact we started to write this book ten years ago. On the occassion of KyotoCGGT2007, we made a special effort to complete this version 1. This book also chronicles the development of mathematical graph theory in Japan, a development which began with many important results in factors and factorizations of graphs. This book has a number of desirable features: 1. It is comprehensive and covers almost all the results from It is self-contained. One who wants to begin research in graph factors and factorizations can confine himself to this one book to follow the history and development of the area, and to find conjectures and open problems. 3. Proof techniques are standardized. Whenever theorems could be proved in a similar way we did so, although different methods of proof were used when they first appeared. 4. Proofs are presented in their entirety for major theorems, even if they are long. We suggest that the readers refer to various parts as needed. 5. Many detailed illustrations are given to accompany the proofs.

8 6. It will be published in electronic form so that it is available to the majority of researchers. We are planning to publish a more polished version 2 adding two more chapters Component Factors and Spanning Trees. So we will appreciate comments from readers. Frank Harary predicted that graph theory will grow so much that each chapter of his book Graph Theory will eventually expand to become a book on its own. He was right. This book is an expansion of his Chapter 9, Factorization. We also predict that the area of factors and factorizations will continue to grow because of many applications to BIBD, Steiner Designs, Matching Theory, OR, etc. that have been found. The following references were very helpful to us during the preparation of this book: Extremal Graph Theory, by Bela Bollobás, Academic Press, (1978) Combinatorial Problems and Exercises,byLászlóLovász, North-Holland, (1979). Matching Theory, by László Lovász and Michael D. Plummer, Annals of Discrete Mathematics 29, North-Holland, (1986). Lutz Volkmann, Regular graphs, regular factors, and the impact of Petersen s Theorems, Jber. d. Dt. Math.-Verein 97 (1995) Lutz Volkmann, Graphen und Digraphen: Eine Einführung in die Graphentheorie (Springer-Verlag) (1991). Mekkia Kouider and Preben D. Vestergaard, Connected factors in graphs A survey, Graphs Combinatorics 21 (2005) Michael D. Plummer, Graph factors and factorization: : A survey, Discrete Mathematics 307 (2007) We would like to thank Haruhide Matsuda, Mari-Jo Ruiz, Lutz Volkmann and our many friends, who work in factor theory, for their assistance in making this book possible. vii Jin Akiyama Mikio Kano

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10 About the Authors Jin Akiyama is Professor of Mathematics and Director of the Research Institute of Educational Development of Tokai University in Tokyo, Japan. His publications are in discrete geometry, graph theory and mathematics education. Mikio Kano is Professor of Mathematics of Ibaraki University in Hitachi, Japan. His research interests include discrete geometry and graph theory. Both authors were involved in the founding of the journal Graphs and Combinatorics and continue to be actively involved in the publication of the journal, Akiyama as Editor-in-Chief and Kano as Managing Editor. ix

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12 Basic Terminology We introduce some graph theory definitions and notation which are needed to discuss factor theory in graphs. A graph G consists of a vertex set V (G) and an edge set E(G). Each edge joins two vertices, which are not necessarily distinct. An edge joining a vertex x to a vertex y is denoted by xy or yx. An edge xy with x = y is called a loop, that is, a loop is an edge joining a vertex to itself. For two distinct vertices, usually there is at most one edge joining them, but if there are two or more edges joining them, then the graph is said to have multiple edges. We can define three types of graphs as follows: A graph that may have loops and multiple edges is called a general graph. A graph that has no loops but may have multiple edges is called a multigraph. A graph having neither loops nor multiple edges is called a simple graph (Figure 1). A graph usually means one of these. The meaning can be understood from the content. There are instances when it is not necessary to distinguish them. In this book, we deal with these three types of graphs and each of them plays an important role. The number of vertices of a graph G is called the order of G and is denoted by G. On the other hand, the number of edges of G is called the size of G and is denoted by G. For a set X, we denote the cardinality of X by X. Then G = V (G) = the order of G, G = E(G) = the size of G. We consider only graphs of finite order, which are called finite graphs. When there exists an edge e = xy joining x to y, we say that e is incident with x and y and vice versa, and x and y are adjacent. Moreover, x and y are called the endvertices of e. If two edges have a common endvertex, then they are said to be adjacent. For a vertex v of a graph G, the number of edges of G incident with v is called the degree of v in G and is denoted by deg G (v). Each loop incident with v contributes two to deg G (v) (Figure 1 (a)). The maximum degree among all vertices of G is called the maximum xi

13 xii 1 5 A loop Multiple edges 4 (a) (b) (c) Figure 1: (a) A general graph with order 7 and size 14, where numbers denote the degrees of the vertices. (b) A multigraph. (c) A simple graph. degree of G and denoted by Δ(G). The minimum degree, denoted by δ(g), of G is defined symmetrically. If every vertex of a graph G has constant degree r, then G is called an r-regular graph. A 3-regular graph is often called an cubic graph. A vertex of odd degree is called an odd vertex, and a vertex of even degree is called an even vertex. Moreover a vertex of degree one is called a leaf or pendant vertex, and the edge incident with an end vertex is called pendant edge. For two given graphs G and H, if there exists a bijection f : V (G) V (H) such that f(x) and f(y) are adjacent in H if and only if x and y are adjacent in G, then we say that G and H are isomorphic and write G = H. The name of the following theorem arises from the fact that if each vertex of a graph represents a person and each edge means that two persons shake hands. Then the sum of degrees is the total number of hands shaken, which is of course equal to twice the number of handshakes. Theorem 1 (Handshaking Theorem) Let G be a general graph. Then the sum of degrees of vertices of G is equal to twice the number of edges of G. Namely, deg G (x) =2 G. x V (G) Proof. Let Ω={(v, e) v V (G),e E(G),v is incident with e.}, which is the set of pairs of vertices and incident edges. Since for every vertex v exactly deg G (v) edges are incident with v, the number of pairs (v, e) inω with fixed v is deg G (v), and since for each edge e exactly two vertices are

14 xiii incident with e, the number of pairs (v, e) in Ω with fixed e is two. Therefore Ω = deg G (x) =2 E(G) =2 G. x V (G) Corollary 2 Every general graph has an even number of odd vertices. Proof. Let G be a general graph, U the set of odd vertices and W the set of even vertices. Then by the Handshaking Theorem, we have 2 G = deg G (x)+ deg G (x) x U x W 0 U (mod 2). Hence U is even. A graph H is called a subgraph of G if V (H) V (G) and E(H) E(G). For a vertex subset X V (G), the subgraph of G whose vertex set is X and whose edge set consists of the edges of G joining vertices of X is called the subgraph of G induced by X and is denoted by X G. A subgraph K of G is called an induced subgraph of G if there exists a vertex subset Y such that K = Y G. We write G X for the subgraph of G induced by V (G) X, which is obtained from G by removing all the vertices in X together with all the edges incident with vertices in X (Figure 2). For an edge set Y E(G), G Y denotes the subgraph of G obtained from G by removing all the edges in Y. A subgraph H of G called a spanning subgraph of G if V (H) =V (G). s y x t X={s,u,x,y} (1) G (2) v u w s y x hxi G u (3) t G - X w Figure 2: (1) A general graph G and its vertex subset X = {s, u, x, y}; induced subgraph X G ; (3) The subgraph G X. (2) The For a given graph H, if a graph G has no induced subgraph isomorphic to H, then we say that G is H-free. The star K 1,3 is often called a claw, and so a K 1,3 -free graph G is often called claw-free.

15 xiv We now give some important classes of simple graphs. Let G be a simple graph. If any two distinct vertices of G are adjacent, then G is called a complete graph. The complete graph of order n is written K n or K(n). If the vertex set of G is partitioned into m disjoint non-empty subsets V (G) =X 1 X 2 X m and any two vertices contained in distinct subsets are adjacent, then G is called a complete m-partite graph. If X i = n i, 1 i m, then such a complete m-partite graph is denoted by K(n 1,n 2,...,n m )ork n1,n 2,...,n m. In particular, K(n 1,n 2 )=K n1,n 2 is called a complete bipartite graph. The complete bipartite graph K(1,n) is called the star of order n + 1. The path P n consists of n vertices v 1,v 2,...,v n and n 1 edges v i v i+1, 1 i n 1, where the two vertices v 1 and v n of degree one are called the endvertices of P n. The cycle C n is obtained from the path P n by adding a new edge joining two endvertices of P n. P 4 K 1,3 K 5 K(2,3,4) K C 4 2,3 =K(2,3) Figure 3: Some graphs. For two graphs G and H, the union G H is the graph with vertex set V (G) V (H) and edge set E(G) E(H), where V (G) V (H) =. The union G G is often denoted by 2G, and for any integer n 3, we inductively define ng by (n 1)G G. The join G + H denotes the graph with vertex V (G) V (H) and edge set E(G + H) =E(G) E(H) {xy x V (G) and y V (H)}, i.e., G + H is obtained from G H by adding all the edges joining a vertex of G to a vertex of H (Figure 4). For a simple graph G, the complement of G, denoted by G, is the graph with vertex set V (G) such that two vertices are adjacent in G if and only if they are non-adjacent in G, in particular, G G = K G. We introduce some notions concerning paths and cycles in a graph. A walk in a graph G is a sequence of vertices and edges (v 0,e 1,v 1,e 2,v 2,...,v n 1,e n,v n ),

16 xv G H G[H 3H G+H _ G Figure 4: The union, the join, and the complement. such that every e i, 1 i n, is an edge joining vertices v i 1 and v i. We say that this walk connects two vertices v 0 and v n. Notice that a walk may pass through some edges and some vertices twice or more. The length of a walk is the number of edges appearing in it, and so the walk given above has length n. A walk that passes through every its edge exactly once is called a trail, and a walk that passes through every one of its vertices exactly once is called a path. It is easy to see that every path is a trail, but a trail is not always a path. We often denote a trail by a sequence (e 1,e 2,...,e n ) of edges and a path by a sequence (v 0,v 1,...,v n ) of vertices. If the initial vertex v 0 and the terminal vertex v n of a walk are the same, then we say that such a walk is closed. A closed trail is called a circuit, and a closed path is called a cycle. A cycle of even length is called an even cycle. Anodd cycle, an even path and an odd path can be defined similarly. a G 1 b h g f 4 c 13 3 d Figure 5: A path (a, b, h, f, d) passing through 1; a cycle (b, h, f, d, c, b) passing through 3; and a circuit (9, 6, 11, 10, 7, 3, 2). A graph G is said to be connected if any two vertices are connected by a path in G. IfG is disconnected, that is, if G is not connected, then G is decomposed into components that are the maximal connected subgraphs of G. A connected simple graph having no cycle is called a tree, and a simple graph having no cycle is called a forest. So each component of a forest is a tree. A forest is called a linear forest if its all components are paths, and

17 xvi is called a star forest if its all components are stars. Theorem 3 Every tree T has the following properties (Figure 6). (i) T = T 1. (ii) For every edge e not contained in T, T + e has a unique cycle, which passes through e. (iii) T has at lest two leaves. e T Figure 6: A tree T with 6 leaves and and an edge e not contained in T. a x b c y G κ(g)=2 and λ(g)=3 X={x,y} T={a,b,c} Figure 7: A 2-connected and 3-edge connected graph G, for which G X and G T are disconnected. A connected graph G, which is not a complete graph, is k-connected if for every vertex set X containing at most k 1 vertices, G X is connected. The connectivity of G is defined to be the maximum k for which G is k-connected, and is denoted by κ(g). Hence if κ(g) = k, then for every X V (G) with X k 1, G X is connected and there exists a subset S V (G) with S = k such that G S is not connected. The connectivity of the complete graph K n is defined to be n 1, that is, κ(k n )=n 1. We can analogously define edge connectivity as follows. A graph G is said to be k-edge connected if for every edge set X containing at most k 1 edges, G X is connected. The edge-connectivity of G is defined to be the maximum k for which G is k-edge connected, and is denote by λ(g) (see Figure 7). The edge-connectivity of the complete graph K n is n 1by

18 xvii the definition. It is easy to see that a connected multigraph G satisfies the following inequality, which was found by Whitney [150]. Equality holds in cubic graphs. κ(g) λ(g) δ(g). Proposition 4 For a cubic multigraph G, it follows that κ(g) =λ(g). The proof of the above lemma is left to the reader (see Figure 8). u u w κ(g)=λ(g)=1 v κ(g)=λ(g)=2 v Figure 8: Connected cubic multigraphs. Let G be a connected graph. Then a vertex v is called a cut vertex if G v is not connected, and an edge e of G is called a bridge or cut edge if G e is not connected. For a disconnected graph, we can similarly define a cut vertex and a bridge. More generally, a vertex subset S V (G) ofa connected graph G is called a vertex cut if G S is disconnected, and an edge subset T E(G) is called a edge cut if G T is disconnected. For two disjoint subsets X and Y of V (G), we denote the set of edges of G joining a vertex of X to a vertex of Y by E G (X, Y ), and the number of edges in E G (X, Y )bye G (X, Y ), i.e., e G (X, Y )= E G (X, Y ) = the number of edges of G joining X to Y. It is easy to see that a minimal edge cut T is expressed as T = E G (X, Y ), where V (G) =X Y, X Y =. A graph G is called a bipartite graph if V (G) is partitioned into two disjoint sets A B so that every edge of G joins a vertex of A to a vertex of B. The two sets A and B are called partite sets and (A, B) is called a bipartition of G. Thus a bipartite graph with bipartition (A, B) can be regarded as a subgraph of K( A, B ). Bipartite graphs plays an important role throughout this book, and the following proposition is useful.

19 xviii Proposition 5 (König [89]) A multigraph G is a bipartite graph if and only if every cycle of G has even length. Proof. Suppose G is a bipartite graph with bipartition (A, B). Then every cycle of G passes alternately through A and B, and thus its length must be even. Conversely, suppose that every cycle of G is of even length. We may assume that G is connected since if G is disconnected, then its components can be considered separately. Assume that G has two distinct vertices such that there are two paths connecting them whose lengths are odd and even, respectively. Choose such a pair (x, y) of vertices so that the sum of lengths of such two paths connecting them is minimum. Let P 1 (x, y) and P 2 (x, y) are such paths connecting x and y, whose length are odd and even respectively. If P 1 (x, y) and P 2 (x, y) meet a vertex z on the way, then either P 1 (x, z) and P 2 (x, z) orp 1 (z, y) and P 2 (z, y) have distinct parities of lengths, which contradicts the choice of (x, y) since the sum of their lengths is shorter than that of P 1 (x, y) and P 2 (x, y) (see Figure 9). Thus P 1 (x, y) and P 2 (x, y) do not meet on the way, which implies that P 1 (x, y)+p 2 (x, y) forms a cycle of odd length. This contradicts the assumption. Therefore, for any two distinct vertices of G, the length of every path connecting them has the same parity. x z y P 1 (x,y) P 2 (x,y) Figure 9: A path P 1 (x, y) of odd length and another path P 2 (x, y) of even length. Choose one vertex v of G, and define two vertex subsets of G as follows: A = {x V (G) There exists a path of even length connecting v to x }, B = {x V (G) There exists a path of odd length connecting v to x }. Then by the above argument, A B =, and it is easy to see that every edge of G joins A to B, and thus G is a bipartite graph with bipartition (A, B). If a graph G can be drawn in the plane so that no two edges intersect except at their endvertices, then G is called an planar graph, and the graph drawn in the plane is called a plane graph (Figure 10). Note that a planar graph can be drawn in different ways as a plane graph, and a plane graph G means one of such plane graphs. For example the two plane G 1 and G 2 in Figure 10 are isomorphic but different plane graphs.

20 xix G 1 G 2 G 3 Figure 10: A plane graph G 1 with five regions, whose vertex degrees are 3,3,5,6 and 7; another plane graph representation G 2 of the same graph; and a maximal planar graph G 3. Given a plane graph G in the plane, a region of G is a maximal portion of the plane in which any two points can be joined by a curve that does not cross any edge of G. The degree of a region R in G, denoted by degree(r), is defined as the number of edges of G in the boundary of R. If a region of G is not a triangle, we can add an edge to G so that the resulting graph is still a plane graph. A planar graph is called a maximal planar graph if we cannot add any new edge without violating the planarity (Figure 10). Hence every region of a maximal planar graph is a triangle, and inversely, a plane graph all of whose regions are triangles is a maximal planar graph. By the same argument as in the proof of Handshaking Theorem, we obtain the following. Lemma 6 For a connected plane graph G, let Ω(G) be the set of regions of G. Then degree(r) =2 G. R Ω(G) The following theorem is very important for planar graphs. Theorem 7 (Euler s Formula) For a connected plane graph G, let p = G, q = G and r be the number of regions of G. Then p q + r =2. Proof. We prove the theorem by induction on G for all connected plane graphs G of fixed order n 2. Since G is connected, a smallest such plane graph is a tree. If G is a tree, then r = 1 and q = p 1 by Theorem 3, and so p q + r = p (p 1) + 1 = 2.

21 xx Thus we may assume that G is not a tree. Then G has a cycle C. Let e be an edge of C. Then G = G e is a connected plane graph, and so by the inductive hypotheses, we have p q r =2, where p = G, q = G and r = the number of regions of G. By substituting p = p, q = q 1 and r = r 1 in the above equation, we have p (q 1) + (r 1) = p q + r =2, which is the desired equation. Hence the theorem is proved. Using the above theorem, we can easily show the following theorem. Theorem 8 Let G be a connected plane simple graph, and let p = G and q = G. Then (i) q 3p 6, where the equality holds if G is a maximal planar graph. (ii) If G has no triangle, in particular, if G is a bipartite plane graph, then q 2p 4. Proof. Let Ω be the set of regions of G. By the lemma, we obtain 3r R Ω degree(r) =2 G =2q. Hence r (2q)/3. By substituting this into Euler s Formula, we have p q + 2q 3 2. Hence 3p 6 q. IfG has no triangle, then every region has degree at least four, and so 4r degree(r) =2 G =2q. R Ω This inequality together with Euler s Formula implies 2p 4 q. A vertex coloring of a graph G is an assignment of positive integers 1, 2,...,k to the vertices of G such that no two adjacent vertices have the same number. Such a vertex coloring is sometimes called a proper vertex coloring. If there exists a vertex coloring with k-colors, then we say G is k-colorable. The minimum number k for which G is k-colorable is called the chromatic number of G and is denoted by χ(g). Analogous definitions for edges can be introduced as follows: An edge coloring of a graph G is an assignment of positive integers 1, 2,...,k to the edges of G such that no

22 two adjacent edges have the same number. Such an edge coloring is said to be proper. We mainly consider proper edge colorings in this book. If there exists a proper edge coloring with k-colors, then we say G is k-edge colorable. The minimum number k for which G is k-edge colorable is called the chromatic index of G and denoted by χ (G). For edge coloring, the following famous theorem holds. Theorem 9 (Vizing [144]) A simple graph G can be properly edge colored with Δ(G)+1 colors. In particular, χ (G) =Δ(G) or Δ(G)+1. We introduced connectivity and edge-connectivity to evaluate the strongness of a graph. There are some other concepts which might evaluate the strongness of a graph in a different point of view. We explain two of them below. The toughness of a connected non-complete graph G, denoted by tough(g), is defined as { tough(g) = min X ω(g X) } ω(g X) 2, X V (G), where ω(g X) denotes the number of components of G X. A graph G is said to be t-tough if tough(g) t. Hence, if G is t-tough, then for a subset S V (G) with ω(g S) 2, it follows that ω(g S) S tough(g) S t. The binding number bind(g) of a graph G, which was introduced by Anderson [12], is defined by { NG (X) } bind(g) = min X V (G), N G (X) V (G). X Thus for every subset S V (G) with N G (S) V (G), it follows that N G (S) bind(g) S. We conclude this introduction with two results on circuits and cycles of graphs. For a graph G, a circuit that passes through every edge of G precisely once is called an Euler circuit of G. A graph having an Euler circuit is called an Eulerian graph. On the other hand, a cycle that passes through every vertex exactly once is called an Hamiltonian cycle. A graph having an Hamiltonian graph is called an Hamiltonian graph. It is easy to see that if a graph G is Hamiltonian, then tough(g) 1. So the non-existence of a Hamiltonian cycle in G 4 in Figure 11 is shown by 3 = ω(g 4 X) > X =2 for a set X, which implies tough(g 4 ) < 1. xxi

23 xxii G 1 G 2 G 3 G 4 Figure 11: An Eulerian multigraph G 1, a non-eulerian graph G 2, a Hamiltonian graph G 3, and a non-hamiltonian graph G 4 with a set X consisting of two bold vertices. Theorem 10 A connected multigraph is Eulerian if and only if every its vertex has even degree. Proof. Assume that G has an Eulerian circuit C. We move along C and remove an edge when we pass through it. Then for every vertex v, when we pass through it, two edges incident with v are removed, and in the final stage, all the edges incident with v are removed, and thus the degree of v is even. Conversely, assume that every vertex of G has even degree. Then we prove that G has an Eulerian circuit by induction on G. If G =2, then G consists of two vertices and two multiple edges joining them, and so G has an Eulerian circuit. Assume that G 3. Since every vertex has degree at least two, G has a cycle C. Every non-trivial component D of G E(C) has vertices of even degrees, where a non-trivial component means a component having at least one edge. Hence by induction, D has an Eulerian circuit Ecir(D). We arrange all the non-trivial components of G E(C) D 1,D 2,...,D m in the order that C passes through at least one of its vertices. Then we can obtain an Eulerian circuit of G as follows: Start at a vertex w of C and go along C to reach a vertex v 1 of D 1, then pass through Ecir(D 1 ) and come back to v 1 and go along C to reach a vertex v 2 of D 2, then pass through Ecir(D 2 ), and continue this procedure until we return to w (see Figure 12 (1)). Then we can obtain a circuit that passes through all the edges of G. We now give a well-known theorem about Hamiltonian cycles, where alpha(g) denotes the cardinality of a maximum independent set of a graph G. Theorem 11 (Chvátal and Erdős, (1972)) Every connected simple graph G with κ(g) α(g) is Hamiltonian unless G = K 2. Proof. Let k = κ(g). Then we may assume k 2 since α(g) = 1 implies

24 xxiii D 1 D 2 v 1 v 2 D 3 D u x 1 y 1 x i y i w C C (1) (2) y j x j Figure 12: (1) An Eulerian multigraph G; (2) a Hamiltonian graph G. G = K n with n 3. Let C be a longest cycle of G, which is considered as a directed cycle. We may assume that C is not Hamiltonian. Since δ(g) κ(g) =k, the length of C is at least k + 1 (see Exercise 2). Let D be a component of G V (C). Since κ(g) =k, there exist at least k distinct vertices in C that are joined to D by edges of G. Denote these vertices by x 1,x 2,...,x k (see Figure 12 (2)). Let y i be the vertex C immediately following x i for all 1 i k. Then y i is not joined to D by an edge of G since otherwise we can easily find a cycle longer than C, which is a contradiction. Moreover, if any two distinct vertices y i and y j are adjacent in G, then we can find a cycle longer than C which starts with a vertex u of D adjacent to x j.itgoestox j, proceeds along C to y i,goestoy j, proceeds along C to x i, goes to a vertex of D adjacent to x i, and then goes to u in D. This contradicts the choice of C. Hence y i and y j are not adjacent in G. Therefore {y 1,y 2,...,y k } {v}, v V (D) is an independent set of G with cardinality k + 1, which contradicts α(g) κ(g) = k. Exercises Exercise 1 Prove that if G is a connected cubic multigraph, then κ(g) = λ(g). Exercise 2 (i) Show that every simple graph with δ(g) 2 has a cycle. (ii) Prove that every simple graph with δ(g) 2 has a cycle with length at least δ(g)+1. Exercise 3 Prove Theorem 3. Exercise 4 Let G be a connected graph and F be a spanning forest of G. Show that the number of components of F is equal to G F. Exercise 5 Let G be a connected simple graph. Show that κ(g) λ(g) δ(g).

25 xxiv Exercise 6 Find a 4-regular simple graph G with κ(g) =2and λ(g) =4. Exercise 7 Let r 2 be an integer. Prove that every connected r-regular bipartite multigraph is 2-edge connected. Exercise 8 Show that Lemma 6 holds. Exercise 9 Prove that every connected graph G satisfies κ(g) 2 tough(g). Exercise 10 Let G be a connected simple graph. Show that if bind(g) > 0, then N G (X) G ( G X )/bind(g) for all X V (G).

26 Chapter 1 Matchings and 1-Factors 1.1 Matchings in Bipartite Graphs In this section we investigate matchings in bipartite graphs and consider results which will play an important role throughout this book. The reader is referred to the book [108] by Lovász and Plummer for a more detailed treatment of matchings. Two edges of a general graph are said to be independent if they have no common endvertex and none of them is a loop. A matching in a general graph G is a set of pairwise independent edges of G (Figure 1.1). In this chapter a bipartite simple graph is briefly called a bipartite graph, and a bipartite graph which may have multiple edges is called a bipartite multigraph. IfM is a matching in a general graph G, then the subgraph of G induced by M, denoted by M G or M, is the subgraph of G whose edge set is M and whose vertex set consists of the vertices incident with some edge in M. In particular, every vertex of M has degree one. Thus it is possible to define a matching as a subgraph whose vertices all have degree one, and we often regard a matching M as its induced subgraph M. Let M be a matching in a general graph G. Then a vertex of G is said to be saturated or covered by M if it is incident with an edge of M; otherwise, it is said to be unsaturated or not covered by M. If every vertex of a vertex subset U of G is saturated by M, then we say that U is saturated by M. A matching with maximum cardinality is called a maximum matching. A matching that saturates all the vertices of G is called a perfect matching or a 1-factor of G (Figure 1.1). It is easy to see that a maximal matching, which is a maximal set of independent edges, is not a maximum matching, and a maximum matching is not a perfect matching. On the other hand, if a matching in a bipartite graph saturates one of its partite sets, then it is 1

27 2 CHAPTER 1. MATCHINGS AND 1-FACTORS M 1 M 2 M 3 Figure 1.1: A maximal matching M 1 ; a maximum matching M 2 ; and a perfect matching M 3. clearly a maximum matching. For a matching M, we write M = the size of M = the number of edges in M, M = the order of M = the number of vertices saturated by M. We first give a criterion for a bipartite multigraph to have a matching that saturates one of its partite sets. Then we apply this criterion to some problems on matchings in bipartite graphs. The criterion mentioned above is given in the following theorem, which was independently found by Hall (1935) and König (1931) and is called the Marriage Theorem. This theorem plays an important role throughout this book, and the proof given here is due to Halmos and Vaughan [54]. An algorithm for finding a maximum matching in a bipartite graph will be given in Algorithm Theorem (Marriage Theorem, Hall [55], König, [90]) Let G be a bipartite multigraph with bipartition (A, B). Then G has a matching that saturates A if and only if N G (S) S for all S A. (1.1) Proof. We first construct a bipartite graph H from the given bipartite multigraph G by replacing all the multiple edges of G by single edges. Then it is obvious that G has the desired matching if and only if H has such a matching, and that G satisfies (1.1) if and only if H satisfies it. Therefore we may assume that G itself has no multiple edges by considering H as the given bipartite graph.

28 1.1. MATCHINGS IN BIPARTITE GRAPHS 3 S NG(S) A M={ } B A G B Figure 1.2: A matching M that saturates A; S and N G (S). Suppose that G has a matching M that saturates A (Figure 1.2). Then for every subset S A, we have N G (S) N M (S) = S. We next prove sufficiency by induction on G. It is clear that we may assume A 2. We consider the following two cases. Case 1. There exists S A such that N G (S) = S. Let H = S N G (S) G and K = (A S) (B N G (S)) G be induced subgraphs of G (Figure 1.3). It is clear that H satisfies condition (1.1), and so H has a matching M H that saturates S by induction. For every subset X A S, we have N K (X) = N G (X S) N G (S) X S S = X. Hence, by induction, K also has a matching M K that saturates A S. Therefore M H M K is the desired matching in G which saturates A. Case 2. N G (S) > S for all S A. Let e = ab (a A, b B) be an edge of G, and let H = G {a, b}. Then for every subset X A {a}, by the assumption of this case, we have N H (X) N G (X) \{b} > X 1, (1.2) which implies N H (X) X. Therefore H has a matching M that saturates A {a} by induction. Then M +e is the desired matching in G. Consequently the theorem is proved. If the edge set E(G) of a multigraph G is partitioned into its disjoint 1-factors E(G) =F 1 F 2 F r, where each F i is a 1-factor of G, then

29 4 CHAPTER 1. MATCHINGS AND 1-FACTORS S NG(S) NG(X) S H NG(S) X A G B X A-S K NK(X) B-NG(S) Figure 1.3: The induced subgraphs H and K. we say that G is 1-factorable, and call this partition a 1-factorization of G. It is trivial that if G is 1-factorable, then G is regular. We now give some results on matchings in bipartite graphs, most of which can be proved by making use of the Marriage Theorem. We begin with the following famous theorem, which was obtained by König in 1916 before the Marriage Theorem. However our proof depends on the Marriage Theorem. Theorem (König [89]) Every regular bipartite multigraph is 1-factorable, in particular, it has a 1-factor (Figure 1.4). Figure 1.4: A 3-regular bipartite multigraph and its 1-factorization. Proof. Let G be an r-regular bipartite multigraph with bipartition (A, B). Then A = B since r A = e G (A, B) =r B. For every subset X A, we have r X = e G (X, N G (X)) r N G (X), and so X N G (X). Hence by the Marriage Theorem, G has a matching M saturating A, which must saturate B since A = B. ThusM is a 1-factor of G. It is obvious that G M is a (r 1)-regular bipartite multigraph, and so it has a 1-factor by the same argument as above. By repeating this procedure, we can obtain a 1-factorization of G.

30 1.1. MATCHINGS IN BIPARTITE GRAPHS 5 Lemma (König [89]) Let G be a bipartite multigraph with maximum degree Δ and let (A, B) be the bipartition of G such that A B. Then there exists a Δ-regular bipartite multigraph which contains G as a subgraph and one of whose bipartite sets is equal to A. a b c d e x y z a b c d e x y z Figure 1.5: A bipartite multigraph and a regular bipartite graph containing it. Proof. By adding A B new vertices to B if A > B, we obtain a bipartite graph with bipartition (A, B ) such that A = B. Add new edges joining vertices in A to vertices in B whose degrees are less than Δ, one at a time until no new edge can be added. We show that by this procedure, we get the desired Δ-regular bipartite multigraph. Suppose that the bipartite multigraph H obtained in this way has a vertex a A such that deg H (a) < Δ. Then there exists a vertex b B such that deg H (b) < Δ because deg H (x) =e H (B,A)= deg H (x) < Δ A =Δ B. x B x A Hence we can add a new edge ab to H, which is a contradiction. Therefore every vertex of A has degree Δ in H, which implies that every vertex of B has degree Δ as A = B. Consequently, H is the desired Δ-regular bipartite multigraph. The next corollary says that the chromatic index of a bipartite multigraph G is equal to the maximum degree of G. Corollary (König [89]) Let G be a bipartite multigraph with maximum degree Δ. Then E(G) can be partitioned into E(G) =E 1 E 2 E Δ such that each E i (1 i Δ) is a matching in G. Proof. By Lemma 1.1.3, there exists a Δ-regular bipartite multigraph H which contains G as a subgraph. Then by Theorem 1.1.2, E(H) can be

31 6 CHAPTER 1. MATCHINGS AND 1-FACTORS partitioned into 1-factors F 1 F 2 F Δ. It is obvious that F i E(G) (1 i Δ) is a matching in G, and their union is equal to E(G). Therefore the corollary holds. We now show some various properties of matchings in bipartite graphs. Some of them are generalizations of the Marriage Theorem and some of them give properties that only bipartite graphs possess. Theorem A bipartite multigraph G has a matching that saturates all the vertices of degree Δ(G). Proof. Let Δ = Δ(G). By Lemma 1.1.3, there exists a Δ-regular bipartite multigraph H which contains G as a subgraph. Then by Theorem 1.1.2, H has a 1-factor F. It is easy to see that F E(G) is a matching of G that saturates all the vertices v of G with degree Δ since every edge of H incident with v is an edge of G. Theorem Let G be a bipartite multigraph with bipartition (A, B). If N G (S) > S for all S A, then for each edge e of G, G has a matching that saturates A and contains e. Proof. Let e = ab (a A, b B) be any edge of G, and H = G {a, b}. Then for every subset X A {a}, we have N H (X) N G (X) \{b} > X 1, which implies N H (X) X. Hence by the Marriage Theorem, H has a matching M saturating A {a}. ThusM + e is the desired matching of G which saturates A and contains e. Theorem Let G be a bipartite graph with bipartition (A, B) such that A = B. Then the following three statements are equivalent. (i) G is connected, and for each edge e, G has a 1-factor containing e. (ii) For every subset X A, N G (X) > X. (iii) For every two vertices a A and b B, G {a, b} has a 1-factor. Proof. (i) (ii) Suppose that N G (Y ) Y for some subset Y A. Since G has a 1-factor F, we have N G (Y ) = Y as N G (Y ) N F (Y ) = Y. Since G is connected, G has an edge e joining a vertex in A Y to a vertex in N G (Y ). However there exists no 1-factor containing e since Y = N G (Y ). This contradicts (i). Thus (ii) holds. (ii) (iii) Let H = G {a, b} and X A {a}. Then N H (X) = N G (X) \{b} > X 1,

32 1.1. MATCHINGS IN BIPARTITE GRAPHS 7 which implies N H (X) X. Hence H has a matching saturating A {a}, which is obviously a 1-factor of H as A {a} = B {b}. (iii) (i) Let e = ab (a A, b B) be an edge of G. Since G {a, b} has a 1-factor F, G has a 1-factor F + e, which contains e. The proof of connectivity will be left to the reader. Theorem Suppose that a bipartite multigraph G with bipartition (A, B) has a matching saturating A. Then there exists a vertex v A possessing the property that every edge incident with v is contained in a matching in G saturating A. Proof. We prove the theorem by induction on A. It is clear that we may assume A 2. If N G (X) > X for all X A, then each vertex in A has the required property by Theorem Hence we may assume that N G (S) S for some S A. Since G has a matching saturating A, it follows from the Marriage Theorem that N G (S) = S. By the inductive hypothesis, the induced subgraph H = S N G (S) G has a vertex v S such that every edge of H incident with v is contained in a matching in H saturating S (Figure 1.6). Note that every edge of G incident with v is contained in H. A B A B v S H N G (S) S N G (S) S = N G (S) M*={ } Figure 1.6: A bipartite multigraph G and its subgraph H; A matching M in G saturating A. Let M be a matching in G saturating A, and M H be a matching in H saturating S. Then it is easy to see that (M \ E(H)) M H is a matching in G that saturates A. Therefore the vertex v in S has the desired property. Proposition Let G be a bipartite multigraph with bipartition (A, B) such that N G (X) X for every X A. If two subsets S, T A satisfy

33 8 CHAPTER 1. MATCHINGS AND 1-FACTORS N G (S) = S and N G (T ) = T, then N G (S T ) = S T and N G (S T ) = S T. In particular, if such subsets exist, there exists a unique maximum subset A 0 A such that N G (A 0 ) = A 0 Proof. Since N G (S T )=N G (S) N G (T ) and N G (S) N G (T ) N G (S T ), we have N G (S T ) = N G (S) + N G (T ) N G (S) N G (T ) S + T N G (S T ) S + T S T = S T. On the other hand, N G (S T ) S T by the assumption. Hence N G (S T ) = S T, and also N G (S T ) = S T by the above inequality. The following theorem is a generalization of the Marriage Theorem since the subgraph H with f(x) = 1 given in the following theorem is nothing but a matching. Theorem (Generalized Marriage Theorem ) Let G be a bipartite graph with bipartition (A, B), and f : A N be a function. Then G has a subgraph H such that if and only if deg H (a) =f(a) for all a A, and (1.3) deg H (b) 1 for all b B (1.4) N G (S) f(x) for all S A. (1.5) x S Proof. We first assume that G has a subgraph H that satisfies (1.3) and (1.4) (Figure 1.7). Then for every S A, we have N G (S) N H (S) = deg H (x) = f(x). x S x S Hence (1.5) holds. In order to prove the sufficiency, we construct a new bipartite graph G with bipartition A B as follows: For every vertex v A, define f(v)

34 1.1. MATCHINGS IN BIPARTITE GRAPHS 9 A* a 3 b 1 c d A 2 1 H={ } G s t u v w x y z a(1) a(2) a(3) b(1) c(1) c(2) d(1) G* s t u v w x y z Figure 1.7: A bipartite graph G and its subgraph H, where numbers denote f(v); and G. vertices v(1),v(2),...,v(f(v)) of A, and connect them to all the vertices of N G (v) by edges (Figure 1.7). Then A = x A f(x) and N G (v(i)) = N G (v) for all v(i) A. For every subset S A, let S = {x A x(i) S for some i}. Then we have by (1.5) N G (S ) = N G (S) x S f(x) S. Hence by the Marriage Theorem, G has a matching M that saturates A. It is clear that the subgraph H of G induced by M satisfies deg H (a) =f(a) for all a A, and deg H (b) 1 for all b B. Hence the theorem is proved. The next theorem gives a formula for the size of maximum matching of a bipartite graph. Moreover, this formula includes the Marriage Theorem as its corollary since N G (S) S for all S A implies M = A as N G ( ) =0. Theorem (Ore [123]) Let G be a bipartite multigraph with bipartition (A, B), and M a maximum matching in G. Then the size of M is given by M = A max S A { S N G(S) }.

35 10 CHAPTER 1. MATCHINGS AND 1-FACTORS Proof. Let M be a maximum matching in G, d = max S A { S N G (S) }, and X A such that X N G (X) = d. Then M = N M (A X) + N M (X) A X + N G (X) = A ( X N G (X) ) = A d. In order to prove the inverse inequality, we construct a new bipartite multigraph H with bipartition (A, B D) from G by adding a new vertex set D of d vertices and by joining every vertex of D to all the vertices of A (Figure 1.8). Then for every S A, since d S N G (S), we have N H (S) = N G (S) + D = N G (S) + d S. Hence H has a matching M H saturating A. Then M H E(G) is a matching in G that contains at least A d edges. Therefore M A d, and the theorem is proved. A S D B N G (S) H Figure 1.8: A new bipartite multigraph H. An induced matching of a general graph G is a matching that is an induced subgraph of G, that is, an induced matching can be expressed as U G for some vertex set U V (G). Theorem (Liu and Xhou, [98]) Let G be a connected bipartite graph with bipartition (A, B). Then the size of a maximum induced matching M of G is given by M = max{ X X A such that N G (Y ) N G (X) for all Y X}. Proof. We say that a subset X A has the property P if N G (Y ) N G (X) for all Y X. Let k = max{ X X A, X has the property P}.

36 1.1. MATCHINGS IN BIPARTITE GRAPHS 11 Let M be a maximum induced matching in G, and let A M and B M be the sets of vertices of A and B, respectively, which are saturated by M. Since M is an induced matching, we have E G (A M,B M )=M. Then for every Y A M, we have N G (Y ) N G (A M ) since N G (Y ) B M = N M (Y ) = Y < A M = N G (A M ) B M. Hence M = A M k. We now show that M k. We may assume k 2. Let S = {a 1,a 2,...,a k } be a maximum subset of A that has the property P and size k. Then N G (Y ) N G (S) for all Y S. For every 1 i k, we have N G (a i ) j i N G (a j )byn G (S \{a i }) N G (S), and so we can choose b i N G (a i ) \ j i N G (a j ). Then {a i b i 1 i k} is an induced matching with k edges, which implies M k. Consequently the theorem is proved. It is remarkable that the following problem is NP-complete [26]: Is there an induced matching of size k in a bipartite graph? For other results, we have polynomial time algorithms, and some of them will be given in Section 1.3. Exercises graph. Recall that a bipartite graph means a bipartite simple Exercise In the proof of Theorem 1.1.1, show that H satisfies condition (1.1). Exercise Prove that if a tree, which is a bipartite graph, has a 1- factor, then it has the unique 1-factor. Exercise Prove that if a tree has a 1-factor, then it consists of the edges e such that the two components of T e are of odd order. Exercise Prove that for every bipartite graph G with maximum degree Δ, there exists a Δ-regular bipartite graph which contains G as a subgraph. Note that (i) both G and H have no multiple edges and (ii) the two bipartite sets of H might be bigger than those of G. Exercise Show that a bipartite graph G has at least Δ(G) distinct maximum matchings, where two matchings M 1 and M 2 are said to be distinct if M 1 M 2.

37 12 CHAPTER 1. MATCHINGS AND 1-FACTORS Exercise Prove the following theorem: Let G be a bipartite graph with bipartition (A, B), and k 1 be an integer. Then G has a spanning subgraph H such that deg H (a) = 1 for all a A, and deg H (b) k for all b B if and only if N G (S) S k for all S A. Exercise Prove that a bipartite graph G has a matching having at least G /Δ(G) edges. 1.2 Covers and Transversals In this section we discuss covers of bipartite graphs and transversals of family of subsets. Some of the results are called min-max theorems because they say that the minimum value of some invariant is equal to the maximum value of another invariant. When a vertex v is incident with an edge e, we say that v covers e and vice versa. A vertex cover of a graph G is a set of vertices that cover all the edges of G. A vertex cover of minimum cardinality is called a minimum vertex cover, and its cardinality is denoted by β(g) (Figure 1.9). Similarly an edge cover of a graph G is defined to be a set of edges that cover all the vertices of G, and an edge cover with minimum cardinality is called a minimum edge cover, and its cardinality is denoted by β (G). Recall that a set of vertices of a graph G is said to be independent if no two of its vertices are adjacent, and a set of edges of G is said to be independent if no two of its edges have an endvertex in common, that is, a set of edges is independent if and only if it forms a matching. We analogously define a maximum independent vertex subset and a maximum independent edge subset of G, and denote their cardinalities by α(g) and α (G), respectively. β(g) = the cardinality of a minimum vertex cover β (G) = the cardinality of a minimum edge cover α(g) = the cardinality of a maximum independent vertex subset α (G) = the cardinality of a maximum independent edge subset

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