On Vizing's edge colouring question

TL;DR

This paper proves that for triangle-free graphs, any proper edge coloring can be transformed into an optimal one through a sequence of Kempe changes, confirming Vizing's question in this case.

Contribution

It establishes that Vizing's question has an affirmative answer specifically for triangle-free graphs, advancing understanding of edge coloring transformations.

Findings

Any proper edge coloring of triangle-free graphs can be converted to an optimal coloring via Kempe changes.

Confirms Vizing's conjecture for a significant class of graphs.

Provides a constructive method for edge coloring adjustments in triangle-free graphs.

Abstract

Soon after his 1964 seminal paper on edge colouring, Vizing asked the following question: can an optimal edge colouring be reached from any given proper edge colouring through a series of Kempe changes? We answer this question in the affirmative for triangle-free graphs.