Sufficient conditions for bipartite graphs to be super-k-restricted edge connected

For a connected graph G = (V, E), an edge set S subset of E is a k-restricted edge cut if G - S is disconnected and every component of G - S contains at least k vertices. The k-restricted edge connectivity of G, denoted by lambda(k)(G), is defined as the cardinality of a minimum k-restricted edge cut. For U-1. U-2 subset of V(G), denote the set of edges of G with one end in U-1 and the other in U-2 by [U-1, U-2]. Define xi k(G) = min] {vertical bar U, V(G)\U vertical bar] : U subset of V(G), vertical bar U vertical bar = k >= 1 and the subgraph induced by U is connected}. A graph G is lambda(k)-optimal i lambda(k)(G) = xi(k)G). Furthermore, if every minimum k-restricted edge cut is a set of edges incident to a certain connected subgraph of order k, then G is said to be super-k-restricted edge connected or super-lambda(k) for simplicity. Let k be a positive integer and let G be a bipartite graph of order n >= 4 with the bipartition (X, Y). In this paper, we prove that: (a) If G has a matching that saturates every vertex in X or every vertex in Y and vertical bar N(u) boolean AND N(v)vertical bar >= 2 for any u, v is an element of X and any u, v is an element of Y, then G is lambda(2)-optimal; (b) If G has a matching that saturates every vertex in X or every vertex in Y and vertical bar N(u) boolean AND N(v)vertical bar >= 3 for any u, v is an element of X and any u, v is an element of Y, then G is super-lambda(2); (c) Ifthe minimum degree delta(G) >= n+2k/4, then G is lambda(k)-optimal; (d) If the minimum degree delta(G) >= n+2k/4, then G is super-lambda(k). (C) 2008 Elsevier B.V. All rights reserved.