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

For a connected graph G = (V , E), an edge set S ⊂ 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 λk(G), is defined as the cardinality of a minimum krestricted edge cut. For U1,U2 ⊂ V (G), denote the set of edges of Gwith one end in U1 and the other in U2 by [U1,U2]. Define ξk(G) = min{|[U, V (G) \ U]| : U ⊂ V (G), |U| = k ≥ 1 and the subgraph induced by U is connected}. A graph G is λk-optimal if λk(G) = ξk(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-λk 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 |N(u)∩N(v)| ≥ 2 for any u, v ∈ X and any u, v ∈ Y , then G is λ2-optimal; (b) If G has a matching that saturates every vertex in X or every vertex in Y and |N(u) ∩ N(v)| ≥ 3 for any u, v ∈ X and any u, v ∈ Y , then G is super-λ2; (c) If theminimumdegree δ(G) ≥ n+2k 4 , then G is λk-optimal; (d) If theminimum degree δ(G) ≥ n+2k+3 4 , then G is super-λk. © 2008 Elsevier B.V. All rights reserved.