On the edge connectivity of direct products with dense graphs

Abstract

Let $κ'(G)$ be the edge connectivity of $G$ and $G\times H$ the direct product of $G$ and $H$. Let $H$ be an arbitrary dense graph with minimal degree $δ(H)>|H|/2$. We prove that for any graph $G$, $κ'(G\times H)=\textup{min}\{2κ'(G)e(H),δ(G)δ(H)\}$, where $e(H)$ denotes the number of edges in $H$. In addition, the structure of minimum edge cuts is described. As an application, we present a necessary and sufficient condition for $G\times K_n(n\ge3)$ to be super edge connected.

8 pages, submited to discrete math