On a kind of restricted edge connectivity of graphs

Abstract

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 λ ( m ) ( G ) is the minimum size of all m -RCs in G . Based on the fact that λ (3) ( G )⩽ ξ 3 ( G ), where ξ m (G)= min {ω(X) : X⊂V,|X|=m and G[X] is connected } ( ω ( X ) denotes the number of edges with one end vertex in X and the other in V ⧹ X ), we call a graph G super- λ (3) if λ (m) (G)=ξ m (G) (1⩽m⩽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- λ (3) . We also characterize super- λ (3) circulant graphs. As a consequence, we give the counting formula for the number of i -cutsets N i of these graphs (including the Star graphs, the Hypercubes and the Harary graphs) for i, 2k−2⩽i<ξ 3 (G) , where k is the regular degree of G .