Extending matchings in claw-free graphs

Abstract

Let G be a graph with a perfect matching and let n be an integer, 1⩽ n<| V( G)|⧸2. Graph G is n- extendable if every matching of size n in G is a subset of a perfect matching. Graph G is bicritical if G— u— v has a perfect matching for every pair of points u and v in V( G). It is proved that every 3-connected claw-free graph is bicritical and for n⩾2, every (2 n+1)-connected claw-free graph is n-extendable. Matching extension in planar and toroidal claw-free graphs is then considered.