Remarks on the Local Irregularity Conjecture

TL;DR

This paper investigates the locally irregular edge coloring of graphs, providing a counterexample to the conjecture that all such graphs are 3-colorable, and confirms the conjecture for specific sparse graph classes.

Contribution

The paper presents a counterexample cactus graph with a chromatic index of 4, challenging the conjecture, and proves the conjecture holds for unicyclic graphs and certain cacti.

Findings

Counterexample cactus graph with CHI'irr = 4

Conjecture holds for unicyclic graphs

Conjecture holds for vertex disjoint cycle cacti

Abstract

A locally irregular graph is a graph in which the end-vertices of every edge have distinct degrees. A locally irregular edge coloring of a graph G is any edge coloring of G such that each of the colors induces a locally irregular subgraph of G. A graph G is colorable if it admits a locally irregular edge coloring. The locally irregular chromatic index of a colorable graph G, denoted by CHI'irr(G), is the smallest number of colors used by a locally irregular edge coloring of G. The Local Irregularity Conjecture claims that all graphs, except odd length path, odd length cycle and a certain class of cacti, are colorable by 3 colors. As the conjecture is valid for graphs with large minimum degree and all non-colorable graphs are vertex disjoint cacti, we take direction to study rather sparse graphs. In this paper, we give a cactus graph B which contradicts this conjecture, i.e. CHI'irr(B) = 4. Nevertheless, we show that the conjecture holds for unicyclic graphs and cacti with vertex disjoint cycles.