On a limit of the method of Tashkinov trees for edge-colouring

TL;DR

This paper identifies a fundamental limitation of the Tashkinov tree method in edge-colouring graphs and introduces a new extension to overcome some of these constraints.

Contribution

It reveals a specific limit of the Tashkinov tree technique and proposes a novel extension to improve its applicability in edge-colouring.

Findings

Identified a limit of the Tashkinov tree method involving Kempe changes

Proposed a new extension to the Tashkinov tree technique

Provides insights into the method's potential and constraints

Abstract

The main technique used to edge-colour graphs requiring $\Delta+2$ or more colours is the method of Tashkinov trees. We present a specific limit to this method, in terms of Kempe changes. We also provide a new Tashkinov tree extension.