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

For a connected graph G, an edge set S is a k-restricted edge-cut if G−S is disconnected and every component of G−S has at least k vertices. Graphs that allow k-restricted edge-cuts are called λk-connected. The k-edge-degree of a graph G is the minimum number of edges between a connected subgraph H of order k and its complement G−H. A λk-connected graph is called λk-optimal if its k-restricted edge-connectivity equals its minimum k-edge-degree and super-λk if every minimum k-restricted edge-cut isolates a connected subgraph of order k. In this paper we consider the cases k=2 and k=3. For triangle-free graphs that are not λk-optimal, we establish lower bounds for the order of components left by a minimum k-restricted edge-cut in terms of the minimum k-edge-degree. Sufficient conditions for a triangle-free graph to be λk-optimal and super-λk follow.