A precolouring extension of Vizing's theorem

TL;DR

This paper extends Vizing's theorem to precoloured edges with a minimum distance, guaranteeing a proper extension of the precolouring to the entire graph under certain conditions, and improves the distance requirement in specific cases.

Contribution

It introduces a new precolouring extension theorem for edge-colouring, confirming a conjecture of Albertson and Moore with a stronger distance condition.

Findings

Extension guaranteed for edges at distance at least 9

Condition reduced to distance 5 for graphs without 5-cycles

First general precolouring extension form of Vizing's theorem

Abstract

Fix a palette $\mathcal K$ of $\Delta+1$ colours, a graph with maximum degree $\Delta$, and a subset $M$ of the edge set with minimum distance between edges at least $9$. If the edges of $M$ are arbitrarily precoloured from $\mathcal K$, then there is guaranteed to be a proper edge-colouring using only colours from $\mathcal K$ that extends the precolouring on $M$ to the entire graph. This result is a first general precolouring extension form of Vizing's theorem, and it proves a conjecture of Albertson and Moore under a slightly stronger distance requirement. We also show that the condition on the distance can be lowered to $5$ when the graph contains no cycle of length $5$.