Local Irregularity Conjecture vs. cacti

TL;DR

This paper investigates the locally irregular chromatic index of cactus graphs, providing evidence that the bow-tie graph B is the unique counterexample to the conjecture that all colorable graphs need at most 3 colors.

Contribution

It proves that all colorable cactus graphs except the bow-tie graph B have a chromatic index of at most 3, supporting the conjecture's validity.

Findings

All colorable cactus graphs except B have X'irr(G) <= 3

B is likely the only counterexample to the conjecture

Supports the conjecture that 3 colors suffice for most graphs

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 locally irregular edge coloring. Recently, it has been observed that the conjecture does not hold for the bow-tie graph B [7]. Cacti are important class of graphs for this conjecture since B and all non-colorable graphs are cacti. In this paper we show that for every colorable cactus graph G != B it holds that X'irr(G) <= 3. This makes us to believe that B is the only colorable graph with X'irr(B) > 3, and consequently that B is the only counterexample to the Local Irregularity Conjecture.