A characterization of the edge connectivity of direct products of graphs

TL;DR

This paper derives a formula for the edge connectivity of the direct product of two graphs and characterizes the structure of their minimum edge-cuts, advancing understanding of graph connectivity in product graphs.

Contribution

It introduces a new function and provides a formula for the edge connectivity of direct product graphs, including structural insights into minimum edge-cuts.

Findings

Derived a formula for edge connectivity of direct product graphs.

Characterized the structure of all minimum edge-cuts.

Established a new function to aid in connectivity analysis.

Abstract

The direct product of graphs $G=(V(G),E(G))$ and $H=(V(H),E(H))$ is the graph, denoted as $G\times H$, with vertex set $V(G\times H)=V(G)\times V(H)$, where vertices $(x_1,y_1)$ and $(x_2,y_2)$ are adjacent in $G\times H$ if $x_1x_2\in E(G)$ and $y_1y_2\in E(H)$. The edge connectivity of a graph $G$, denoted as $\lambda(G)$, is the size of a minimum edge-cut in $G$. We introduce a function $\psi$ and prove the following formula %for the edge-connectivity of direct products $$\lambda (G\times H)=\min {2\lambda(G)|E(H)|,2\lambda(H)|E(G)|,\delta(G\times H), \psi(G,H), \psi(H,G)} .$$ We also describe the structure of every minimum edge-cut in $G\times H$.