Extension from Precoloured Sets of Edges

Abstract

We consider precolouring extension problems for proper edge-colourings of graphs and multigraphs, in an attempt to prove stronger versions of Vizing's and Shannon's bounds on the chromatic index of (multi)graphs in terms of their maximum degree $\Delta$. We are especially interested in the following question: when is it possible to extend a precoloured matching to a colouring of all edges of a (multi)graph? This question turns out to be related to the notorious List Colouring Conjecture and other classic notions of choosability.

26 pages, 3 figures, 1 table; in v2, two co-authors added, main conjecture weakened, weak version of conjecture proved, planar section updated; v3 accepted to Electronic Journal of Combinatorics