On super 2-restricted and 3-restricted edge-connected vertex transitive graphs

Let G = (V (G), E(G)) be a simple connected graph and F subset of E(G). An edge set F is an m-restricted edge cut if G - F is disconnected and each component of G - F contains at least m vertices. Let lambda((m))(G) be the minimum size of all m-restricted edge cuts and xi(m)(G) = min{vertical bar omega(U)vertical bar : vertical bar U vertical bar = m and G[U] is connected}, where omega(U) is the set of edges with exactly one end vertex in U and G[U] is the subgraph of G induced by U. A graph G is optimal-lambda((m)) if lambda((m))(G) = xi(m)(G). An optimal-lambda((m)) graph is called super m-restricted edge-connected if every minimum m-restricted edge cut is omega(U) for some vertex set U with vertical bar U vertical bar = m and G[U] being connected. In this note, we give a characterization of super 2-restricted edge-connected vertex transitive graphs and obtain a sharp sufficient condition for an optimal-lambda((3)) vertex transitive graph to be super 3-restricted edge-connected. In particular, a complete characterization for an optimal-lambda((2)) minimal Cayley graph to be super 2-restricted edge-connected is obtained. (C) 2011 Elsevier B.V. All rights reserved.