On the edge connectivity of direct products with dense graphs

TL;DR

This paper establishes a formula for the edge connectivity of the direct product of any graph with a dense graph having a high minimum degree, and characterizes the structure of minimum edge cuts.

Contribution

It provides a new exact formula for the edge connectivity of direct products with dense graphs and describes the structure of their minimum edge cuts.

Findings

Derived the exact edge connectivity formula for G×H with dense H

Characterized the structure of minimum edge cuts in such products

Identified conditions for super edge connectivity of G×K_n

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.