A note on the locally irregular edge colorings of cacti

TL;DR

This paper investigates the locally irregular edge coloring of cactus graphs, establishing that all colorable cactus graphs require at most four colors, thus providing a partial validation of the Local Irregularity Conjecture.

Contribution

It proves that the locally irregular chromatic index of any colorable cactus graph is at most four, extending understanding of the conjecture to this class of graphs.

Findings

All colorable cactus graphs have a locally irregular chromatic index of at most 4.

The conjecture that all such graphs require at most 3 colors is not universally true.

The result supports the conjecture for a broader class of graphs, namely cacti.

Abstract

A graph is locally irregular if the degrees of the end-vertices of every edge are distinct. An edge coloring of a graph G is locally irregular if every color induces a locally irregular subgraph of G. A colorable graph G is any graph which admits a locally irregular edge coloring. The locally irregular chromatic index X'irr(G) of a colorable graph G is the smallest number of colors required by a locally irregular edge coloring of G. The Local Irregularity Conjecture claims that all colorable graphs require at most 3 colors for a locally irregular edge coloring. Recently, it has been observed that the conjecture does not hold for the bow-tie graph B, since B is colorable and requires at least 4 colors for a locally irregular edge coloring. Since B is a cactus graph and all non-colorable graphs are also cacti, this seems to be a relevant class of graphs for the Local Irregularity Conjecture. In this paper we establish that X'irr(G)<= 4 for all colorable cactus graphs.