Journal of Graph Theory
Matching extension and minimum degree
Discrete Mathematics
On the structure of minimally n-extendable bipartite graphs
Discrete Mathematics
Graph Theory With Applications
Graph Theory With Applications
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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.