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 $n\ge 2k$ . In this paper, we prove that if the minimum degree $\delta (G)\ge k+1-{(-1)}^k$ and there are at least $k+\frac{1+{(‐1)}^k}{2}$ common vertices in the neighbor sets of each pair of nonadjacent vertices in G, then G is super k‐restricted edge‐connected.