On the super connectivity of Kronecker products of graphs

Abstract

Let G1 and G2 be two graphs. The Kronecker productG1×G2 has vertex set V(G1×G2)=V(G1)×V(G2) and edge set E(G1×G2)={(u1,v1)(u2,v2):u1u2∈E(G1) and v1v2∈E(G2)}. A graph G is super connected, or simply super-κ, if every minimum separating set is the neighbors of a vertex of G, that is, every minimum separating set isolates a vertex. In this paper we show that if G is a graph with κ(G)=δ(G) and Kn(n⩾3) a complete graph on n vertices, except that G is a complete bipartite graph Km,m (m⩾1) and Kn=K3, then G×Kn is super-κ, where κ(G) and δ(G) are the connectivity and the minimum degree of G, respectively.