Precoloring extension of Vizing’s Theorem for multigraphs

Abstract

Let $G$ be a graph with maximum degree $\Delta \left(G\right)$ and maximum multiplicity $\mu \left(G\right)$. Vizing and Gupta, independently, proved in the 1960s that the chromatic index of $G$ is at most $\Delta \left(G\right)+\mu \left(G\right)$. 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 $t$. Albertson and Moore conjectured that if $G$ is a simple graph, using the palette $\{1,\dots ,\Delta \left(G\right)+1\}$, any precoloring on a distance-3 matching can be extended to a proper edge coloring of $G$. Edwards et al. proposed the following stronger conjecture: For any graph $G$, using the palette $\{1,\dots ,\Delta \left(G\right)+\mu \left(G\right)\}$, any precoloring on a distance-2 matching can be extended to a proper edge coloring of $G$. Girão and Kang verified the conjecture of Edwards et al. for distance-9 matchings. In this paper, we improve the required distance from 9 to 3 for multigraphs $G$ with $\mu \left(G\right)\ge 2$.