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## On the edge connectivity of direct products with dense graphs

### Abstract

Let $\kappa'(G)$ 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 $G$, $\kappa'(G\times H)=\textup{min}\{2\kappa'(G)e(H),\delta(G)\delta(H)\}$, 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 $G\times K_n(n\ge3)$ to be super edge connected.Comment: 8 pages, submited to discrete mat

Topics: Mathematics - Combinatorics
Year: 2011
OAI identifier: oai:arXiv.org:1102.5181