Precoloring extension of Vizing's Theorem for multigraphs

TL;DR

This paper improves the conditions under which a precoloring of a distance-2 matching in a multigraph can be extended to a proper edge coloring, reducing the required distance from 9 to 3.

Contribution

It advances the understanding of precoloring extension in multigraphs by lowering the distance requirement from 9 to 3 for certain graphs.

Findings

Precoloring extension is possible for distance-3 matchings in multigraphs with multiplicity at least 2.

The result narrows the gap towards Edwards et al.'s conjecture for distance-2 matchings.

Improves previous bounds, contributing to the theory of edge colorings in multigraphs.

Abstract

Let $G$ be a graph with maximum degree $\Delta(G)$ and maximum multiplicity $\mu(G)$. Vizing and Gupta, independently, proved in the 1960s that the chromatic index of $G$ is at most $\Delta(G)+\mu(G)$. The distance between two edges $e$ and $f$ in $G$ is the length of a shortest path connecting an endvertex of $e$ and an endvertex of $f$. A distance-$t$ matching is a set of edges having pairwise distance at least $t$. Edwards et al. proposed the following conjecture: For any graph $G$, using the palette $\{1, \dots, \Delta(G)+\mu(G)\}$, any precoloring on a distance-$2$ matching can be extended to a proper edge coloring of $G$. Gir\~{a}o and Kang verified this conjecture for distance-$9$ matchings. In this paper, we improve the required distance from $9$ to $3$ for multigraphs $G$ with $\mu(G) \ge 2$.