On restricted edge-connectivity of replacement product graphs

TL;DR

This paper investigates the edge-connectivity and restricted edge-connectivity of replacement product graphs, providing bounds, exact values for special cases, and constructing Cayley graphs with specific restricted edge-connectivity properties, solving a decade-old problem.

Contribution

It establishes bounds and exact values for edge-connectivity of replacement product graphs, characterizes when their Cayley graph products are Cayley graphs, and constructs graphs with prescribed restricted edge-connectivity.

Findings

Derived bounds on edge-connectivity and restricted edge-connectivity.

Identified conditions for Cayley graph replacement products to remain Cayley graphs.

Constructed Cayley graphs with maximum restricted edge-connectivity equal to degree plus s.

Abstract

This paper considers the edge-connectivity and restricted edge-connectivity of replacement product graphs, gives some bounds on edge-connectivity and restricted edge-connectivity of replacement product graphs and determines the exact values for some special graphs. In particular, the authors further confirm that under certain conditions, the replacement product of two Cayley graphs is also a Cayley graph, and give a necessary and sufficient condition for such Cayley graphs to have maximum restricted edge-connectivity. Based on these results, the authors construct a Cayley graph with degree $d$ whose restricted edge-connectivity is equal to $d+s$ for given odd integer $d$ and integer $s$ with $d \geqslant 5$ and $1\leqslant s\leqslant d-3$, which answers a problem proposed ten years ago.