On an Edge Precoloring Conjecture

TL;DR

This paper investigates a conjecture related to edge coloring in graphs, providing counterexamples to a proposed strengthening of Vizing's Theorem and establishing a weaker, more attainable version.

Contribution

The authors present an infinite family of counterexamples to the conjectured edge precoloring extension and prove a weaker, more feasible version of the conjecture.

Findings

Counterexamples disprove the conjecture.

A weaker version of the conjecture is proven.

Insights into edge coloring extensions in graphs.

Abstract

Edwards, van den Heuvel, Kang, and Sereni conjectured the following strengthening of Vizing's Theorem: let $G$ be a simple graph, and let $K = \Delta(G) + 1$. For any matching $M$ in $G$ and any precoloring of the edges in $M$ using the colors $\{1, \ldots, K\}$, there is some proper $K$-edge-coloring of $G$ extending the given precoloring. We give an infinite family of counterexamples to this conjecture, and prove a weaker version of the conjecture proposed in the same work.