Matching extensions of strongly regular graphs

Let J3 be the number of vertices commonly adjacent to any pair of non-adjacent vertices. It is proved that every strongly regular graph with even order and J3 ~ 1 is l-extendable. We also show that every strongly regular graph of degree at least 3 and cyclic edge connecti vity at least 3k -3 is 2-extendab Ie. Strongly regular graphs of k even order and of degree k at least 3 with J3 ~"3 are 2-extendab Ie, except the Petersen graph and one other graph. 1. I nt rod uct ion a nd term i nolog y All graphs considered are finite, undirected. connected and simple. A graph G is called strongly regular if G is k-regular and there are two integers' eX,~ ~ 0 such that for each pair of vertices u and v, u ;z: v, the number of the common neighbours of u and v is eX or ~ according as u and v are adjacent or non-adjacent. A strongly regular graph with v vertices is called a (v, k. eX, J3)-graph. These general parameters will be assumed unless stated otherw ise. A graph G is called cycl ically m-edge-connected if I S I ~ m for each edge cutset S of G such that there are two components in G S each of which has a cycle. The set S is called a cyclic edge cutset. The size of a minimal cyclic edge cutset is called the cyclic edge connect! vi ty. and is denoted by c:A(G). Australasian Journal of Combinatorics §( 1992), pp.187-208 Suppose G has a perfect matching. A graph G is called n-extendable if for the given integer n ~ (v-2)/2. G has n independent edges and any n independent edges are contained in a perfect matching of G. In [1 J. the n-extendabi Ii ty of edge (vertex) transi ti ve graphs is discussed. When the cyclic edge connectivity is large enough. an edge transitive graph is n-extendable. We show here that there is a similar relation between cyclic edge connectivity and n-extendability in strongly regular graphs. We also find some n-extendable strongly regular graphs for arb i trary n. All terminology and notation not defined in this paper can be found in [2 J. Reference [4J provides a strong background for matching theory and [6] gives a survey of results in strongly regular graphs. Lemma If G is a strongly regular graph. then k(k-o(-l) = (v-k-l )~. Proof See Theorem 2.2 in [3]. o If G is a cubic strongly regular graph. then by Lemma 1. 3(2-0() = (v-4)~. For 0( = 2. we find G = K... If 0( = 1. then v is odd which is not possible since G is cubic. If 0( = O. then ~ = 1 and v = 10 or ~ = 3 and v = 6. For ~ = 1. we obtain the Petersen graph P. whi Ie for ~ = 3 we get KJ,J' Lemma 2 Let G be a graph with even order. G is n-extendable. Proof See [5]. 2. Matching of strongly regular graphs v If 8(G) > + n. then -2