Connectivity of k-extendable graphs with large k Original Research Article

Let G be a simple connected graph on 2n vertices with perfect matching. For a given positive integer k (0⩽k⩽n−1), G is k-extendable if any matching of size k in G is contained in a perfect matching of G. It is proved that if G is a k-extendable graph on 2n vertices with k⩾n/2, then either G is bipartite or the connectivity of G is at least 2k. As a corollary, we show that if G is a maximal k-extendable graph on 2n vertices with n+2⩽2k+1, then G is Kn,n if k+1⩽δ⩽n and G is K2n if 2k+1⩽δ⩽2n−1. Moreover, if G is a minimal k-extendable graph on 2n vertices with n+1⩽2k+1 and k+1⩽δ⩽n then the minimum degree of G is k+1. We also discuss the relationship between the k-extendable graphs and the Hamiltonian graphs.