On a kind of restricted edge connectivity of graphs

Let G=(V; E) be a connected graph and S ⊂E. S is said to be a m-restricted edge cut (m-RC) if G − S is disconnected and each component contains at least m vertices. The m-restricted edge connectivity (G) is the minimum size of all m-RCs in G. Based on the fact that (G)6 3(G), where m(G)=min{!(X ): X ⊂V; |X |=m and G[X ] is connected} (!(X ) denotes the number of edges with one end vertex in X and the other in V\X ), we call a graph G super(3) if (G)= m(G) (16m6 3). We proved that regular graphs with order more than 5 have at least one 3-RC, and show that vertex-and edge-transitive graphs other than cycles are super. We also characterize super(3) circulant graphs. As a consequence, we give the counting formula for the number of i-cutsets Ni of these graphs (including the Star graphs, the Hypercubes and the Harary graphs) for i; 2k − 26 i¡ 3(G), where k is the regular degree of G. ? 2002 Elsevier Science B.V. All rights reserved.