The palette index of the Cartesian product of paths, cycles and regular graphs

TL;DR

This paper investigates the palette index of Cartesian products of graphs, especially paths, cycles, and regular graphs, providing new bounds and exact values for these graph classes.

Contribution

It offers new bounds and exact results on the palette index for Cartesian products involving paths, cycles, and regular or nearly regular graphs.

Findings

Exact palette index values for certain graph products

Bounds on palette index for regular and nearly regular graph products

Insights into how graph structure affects palette diversity

Abstract

The palette of a vertex v in a graph G is the set of colors assigned to the edges incident to v. The palette index of G is the minimum number of distinct palettes among the vertices, taken over all proper edge colorings of G. This paper presents results on the palette index of the Cartesian product $G \Box H$, where one of the factor graphs is a path or a cycle. Additionally, it provides exact results and bounds on the palette index of the Cartesian product of two graphs, where one factor graph is isomorphic to a regular or class 1 nearly regular graph.