Strong edge coloring of Cayley graphs and some product graphs

TL;DR

This paper determines the exact strong chromatic index of unitary Cayley graphs, explores their product structure, and provides tight bounds for the strong chromatic index of Cartesian products of trees and cycles, including exact formulas in specific cases.

Contribution

It establishes the exact strong chromatic index for all unitary Cayley graphs and derives tight bounds for the strong chromatic index of product graphs involving trees and cycles.

Findings

Exact strong chromatic index for all unitary Cayley graphs.

Tight bounds for Cartesian products of trees and cycles.

Exact formulas for products involving stars and cycles divisible by 4.

Abstract

A strong edge coloring of a graph $G$ is a proper edge coloring of $G$ such that every color class is an induced matching. The minimum number of colors required is termed the strong chromatic index. In this paper, we determine the exact value of the strong chromatic index of all unitary Cayley graphs. Our investigations reveal an underlying product structure from which the unitary Cayley graphs emerge. We then go on to give tight bounds for the strong chromatic index of the Cartesian product of two trees, including an exact formula for the product in the case of stars. Further, we give bounds for the strong chromatic index of the product of a tree with a cycle. For any tree, those bounds may differ from the actual value only by not more than a small additive constant (at most 2 for even cycles and at most 5 for odd cycles), moreover they yield the exact value when the length of the cycle is divisible by $4$.