Matchings and matching extensions in graphs

Abstract Let G be a graph with a perfect matching and k be an integer such that 1 ⩽ k vbV ( G ) vb /2. Then G is said to be k -extendable if every matching of size k in G extends to a perfect matching of G . Plummer (1994) proved that every (2 k + 1)-connected K 1,3 -free graph of even order is k -extendable. In this paper, it was proved that every (2 k + n − 2)-connected K 1, n -free graph of even order is k -extendable. Also, an answer to the problem of characterizing maximal k -extendable graphs posted by Plummer (1994) and Saito (1989/90) is given. Besides, we show that a regular graph of even order belongs to the first class if its any two odd cycles have at least a vertex in common.