Cartesian Product and Acyclic Edge Colouring

TL;DR

This paper establishes an upper bound on the acyclic chromatic index of the Cartesian product of two graphs, extending previous results on grid-like graphs and contributing to graph coloring theory.

Contribution

It proves that the acyclic chromatic index of the Cartesian product of two graphs is at most the sum of their individual indices, generalizing prior specific cases.

Findings

Proved $a'(G ox H) \,\le\, a'(G) + a'(H)$ for graphs with max index > 1.

Extended bounds from grid-like graphs to general Cartesian products.

Provides a constructive approach for acyclic edge coloring of product graphs.

Abstract

The acyclic chromatic index, denoted by $a'(G)$, of a graph $G$ is the minimum number of colours used in any proper edge colouring of $G$ such that the union of any two colour classes does not contain a cycle, that is, forms a forest. We show that $a'(G\Box H)\le a'(G) + a'(H)$ for any two graphs $G$ and $H$ such that $max\{a'(G), a'(H)\} > 1$. Here, $G \Box H$ denotes the cartesian product of $G$ and $H$. This extends a recent result of [15] where tight and constructive bounds on $a'(G)$ were obtained for a class of grid-like graphs which can be expressed as the cartesian product of a number of paths and cycles.