Matching extension and minimum degree

Let G be a simple connected graph on 2n vertices with a perfect matching. For a given positive integer k, 1 k n ? 1, G is k-extendable if for every matching M of size k in G, there exists a perfect matching in G containing all the edges of M. The problem that arises is that of characterizing k-extendable graphs. In this paper, we establish a necessary condition, in terms of minimum degree, for k-extendable graphs. Further, we determine the set of realizable values for minimum degree of k-extendable graphs. In addition, we establish some results on bipartite graphs including a sufficient condition for a bipartite graph to be k-extendable.