Precoloring extension of Vizing's Theorem for 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 $G$. Girão 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 $\mu (G)\ge 2$.

23 pages,4 figures