On an Edge Precoloring Conjecture

Edwards, van den Heuvel, Kang, and Sereni conjectured the following strengthening of Vizing’s Theorem: let $G$ be a simple graph, and let $K = \Delta(G) + 1$\mathrm{.}Foranymatching$M$in$G$andanyprecoloringoftheedgesin$M$usingthecolors${1, \ldots, K}$,thereissomeproper$K$-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.