Extension from Precoloured Sets of Edges

TL;DR

This paper investigates conditions under which precoloured edges in graphs can be extended to full proper edge-colourings, linking the problem to longstanding conjectures like the List Colouring Conjecture and aiming to strengthen bounds on the chromatic index.

Contribution

It introduces new insights into precolouring extension problems for edge-colourings, connecting them to classic conjectures and bounds in graph theory.

Findings

Identifies conditions for extending precoloured matchings to full colourings.

Establishes connections between extension problems and the List Colouring Conjecture.

Provides potential pathways to strengthen Vizing's and Shannon's bounds.

Abstract

We consider precolouring extension problems for proper edge-colourings of graphs and multigraphs, in an attempt to prove stronger versions of Vizing's and Shannon's bounds on the chromatic index of (multi)graphs in terms of their maximum degree $\Delta$. We are especially interested in the following question: when is it possible to extend a precoloured matching to a colouring of all edges of a (multi)graph? This question turns out to be related to the notorious List Colouring Conjecture and other classic notions of choosability.