On super restricted edge-connectivity of vertex-transitive graphs

Let X = (V, E) be a connected vertex-transitive graph with degree k. Call X super restricted edge-connected, in short, sup-lambda', if F is a minimum edge set of X such that X - F is disconnected and every component of X - F has at least two vertices, then F is the set of edges adjacent to a certain edge in X. Wang [Y, Q, Wang, Super restricted edge-connectivity of vertex-transitive graphs, Discrete Mathematics 289 (2004) 199-205] proved that a connected vertex-transitive graph with degree k > 2 and girth g > 4 is sup-lambda'. In this paper, by studying the lambda'-superatom of X, we present sufficient and necessary conditions for connected vertex-transitive graphs and Cayley graphs with degree k > 2 to be sup-lambda'. In particular, sup-lambda' connected vertex-transitive graphs with degree k > 2 and girth g > 3 are completely characterized. These results can be seen as an improvement of the one which is obtained by Wang.