Sufficient conditions for bipartite graphs to be super- [formula omitted]-restricted edge connected

Abstract

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 λ k ( G ) , is defined as the cardinality of a minimum k -restricted edge cut. For U 1 , U 2 ⊂ 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 ξ k ( G ) = min { | [ U , V ( G ) ∖ U ] | : U ⊂ V ( G ) , | U | = k ≥ 1 and the subgraph induced by U is connected } . A graph G is λ k -optimal if λ k ( G ) = ξ 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- λ 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 | N ( u ) ∩ N ( v ) | ≥ 2 for any u , v ∈ X and any u , v ∈ Y , then G is λ 2 -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- λ 2 ; (c) If the minimum degree δ ( G ) ≥ n + 2 k 4 , then G is λ k -optimal; (d) If the minimum degree δ ( G ) ≥ n + 2 k + 3 4 , then G is super- λ k .