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

An edge cut S of a connected graph G = (V, E) is a k-restricted edge cut if every component of G S contains at least k vertices. A graph is said to be super k-restricted edge-connected if every minimum k-restricted edge cut is a set of edges incident to a certain connected subgraph of order k. Let k be a positive integer, and let G be a connected triangle-free graph of order n >= 2k. In this paper, we prove that if the minimum degree delta(G) >= k + 1 (-1)(k) and there are at least k + 1+(-1)(k)/2 common vertices in the neighbor sets of each pair of nonadjacent vertices in G, then G is super k-restricted edge-connected. (C) 2015 Elsevier B.V. All rights reserved.