A note on transformations of edge colorings of chordless graphs and triangle-free graphs

TL;DR

This paper extends a known result about transforming edge colorings via Kempe changes from triangle-free graphs to a broader class including chordless graphs, showing similar transformation capabilities.

Contribution

It introduces a modification of existing proof techniques to include chordless graphs alongside triangle-free graphs for edge coloring transformations.

Findings

Optimal edge coloring can be transformed via Kempe changes in triangle-free graphs.

The method extends to chordless graphs, broadening applicability.

The proof modification is small but effective for larger graph classes.

Abstract

Bonamy et al. (2023) proved that an optimal edge coloring of a simple triangle--free graph $G$ can be reached from any given proper edge coloring of $G$ through a series of Kempe changes. We show that a small modification of their proof gives a possibility to obtain a similar result for a larger class of simple graphs consisting of all triangle-free and all chordless graphs (a graph $G$ is chordless if in every cycle $C$ of $G$ any two nonconsecutive vertices of $C$ are not adjacent).