On matchability of graphs

A graph is h-matchable if G-X has a perfect matching for every subset X ~ V(G) with IXI = h, and it is h-extendable if every matching of h edges can be extended to a perfect matching. It is proved that a graph G with even order is 2h-matchable if and only if (1) G is h-extendable; and (2) for any edge set D such that, for each e = xy E D, x,y E V(G) and e ~ E( G), G U D is h-extendable. Also nine known sufficient conditions for a graph to be h-extendable are stated, and sharp analogues of them all are obtained for matchability, each of which implies the corresponding result for extendability. 1 Terminology and introduction All graphs considered in this paper are undirected, finite and simple. In general we follow the terminology of [1]. Let G be a graph. We denote by o(G) the number of odd components of G and by w(G) the number of the components of G. Let v E V(G) and X ~ V(G). We define N(v) = {u I u E V(G) and uv E E(G)} and N(X) = U N(v). Let S ~ V(G) vEX and let H be a subgraph of G. We ~se the notation Ns(v) = N(v) n S, NH(v) = N(v) n V(H), ds(v) = INs(v)1 and dH(v) = INH(v)l. Let G and H be two disjoint graphs. We denote by kH the union of k copies of Hi and by G+ H the join of G and H, which is the graph constructed from G and H by joining each vertex of G to all vertices of H. A graph G with n vertices is h-matchable where 0 ::; h :::; n-2, if for each subset X ~ V(G) with IXI = h, G-X has a perfect matching (a I-factor). When h = 0, G has a perfect matching. When h = 1 or 2, G is known as factor-critical or bicritical respectively. G is h-extendable for 0 ::; h :::; (n-2)/2 if G has a matching Australasian Journal of Combinatorics 21(2000), pp.201-210 of size h and any matching of size h in G is contained in a perfect matching of G. When h = 0, G has a perfect matching. The toughness of G is defined as: tough(G) = min { W(~IX) I X C V(G) and w(G-X) ~ 2 } if G is not a complete graph, and tough(G) = 00 if G is a complete graph. The binding number of G is defined as: bind(G) = min { I~~)I I 0 =1= X C V(G) and N(X) =1= V(G) }. The concept of h-extendability was introduced by Plummer [6] in 1980. Since then, several general sufficient conditions for h-extendability have been found (see [2], [4-8] and Section 4 below). For each of these conditions, we shall obtain an analogous sharp sufficient condition for a graph to be h-matchable, and we shall see in Section 4 that each of our new theorems implies the corresponding result for extend ability. Also we shall obtain a result to show the relation between matchability and extend ability in Section 2. 2 A few properties of h-matchable graphs In this Section, we show some important properties of h-matchable graphs of which we shall make frequent use in the next section. Proposition 1: Let G be a graph with order nand h be an integer such that 0 ::; h ::; n-2 and h == n (mod 2). Then G is h-matchable if and only if, for each subset S ~ V(G) with lSI ~ h, o(G-S) ::; lSI h. Proof. This follows easily from Tutte's well known characterization of perfect matchings [11] (see also [10], Theorem 3.3.12.) 0 Corollary 2: Let G be anh-matchable graph. Then G is j-matchable for every j such that 0 ::; j ::; hand j == h (mod 2). Proof. We use Proposition 1. Suppose S ~ V(G) and lSI ~ j. If j ::; lSI < h, then SeT for some set T such that ITI = h, and then o(G-S) ::; o(G-T) + (h-j) ::; (lSI-h) + (h-j) = ISI-j; this holds because removing a vertex from a graph cannot reduce its number of odd components by more than 1. If lSI ~ h then o(G-S) ::;