Minimum degree of minimal defect n-extendable bipartite graphs

A near perfect matching is a matching saturating all but one vertex in a graph. If G is a connected graph and any n independent edges in G are contained in a near perfect matching, then G is said to be defect n-extendable. If for any edge e in a defect n-extendable graph G, G-e is not defect n-extendable, then G is minimal defect n-extendable. The minimum degree and the connectivity of a graph G are denoted by delta(G) and kappa(G) respectively. In this paper, we study the minimum degree of minimal defect n-extendable bipartite graphs. We prove that a minimal defect 1-extendable bipartite graph G has delta(G) = 1. Consider a minimal defect n-extendable bipartite graph G with n >= 2, we show that if kappa(G) = 1, then delta(G) <= n + 1 and if kappa(G) >= 2, then 2 <= delta(G) = kappa(G) <= n + 1. In addition, graphs are also constructed showing that, in all cases but one, there exist graphs with minimum degree that satisfies the established bounds. (C) 2009 Elsevier B.V. All rights reserved.