On the edge connectivity of direct products with dense graphs

Abstract

Let ${\kappa }^{\prime }(G)$ 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 $\delta (H)>|H|/2$. We prove that for any graph $G$, ${\kappa }^{\prime }(G\times H)=\text{min}\{2{\kappa }^{\prime }(G)e(H),\delta (G)\delta (H)\}$, 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 $G\times K_n(n\ge 3)$ to be super edge connected.

8 pages, submited to discrete math