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

Abstract

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 δ(G)≥k+1−(−1)k and there are at least k+1+(−1)k2 common vertices in the neighbor sets of each pair of nonadjacent vertices in G, then G is super k-restricted edge-connected.