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.
Highlights
Let G = (V, E) be agraph and let K = [K] = {1, . . . , K} be a palette of available colours. (In this paper, a multigraph can have multiple edges, but no loops; while a graph is always simple.) We consider the following question: given a subset S ⊆ E of edges and a proper colouring of elements of S using only colours from K, is there a proper colouring of all edges of G in concordance with the given colouring on S? We may consider the set S as a set of precoloured edges, while the full colouring, if it exists, may be considered as extending the precolouring
Due to the remarkable work of Kahn [22, 23, 24] on edge-colourings and list edgecolourings ofgraphs, does an asymptotic form of Conjecture 3 hold, but so does a precolouring extension of an asymptotic form of the Goldberg–Seymour Conjecture
It appears that the List Colouring Conjecture (LCC) and our conjecture are independent statements, we have obtained several results corresponding to specific areas of success in list edge-colouring
Summary
1] conjectured that when G is a simple graph, any precoloured distance-3 matching can be extended to a proper edge-colouring of G using the palette K = [∆(G) + 1]. Using the palette K = [∆(G) + μ(G)], any precoloured distance-2 matching can be extended to a proper edge-colouring of all of G. We prove several special cases of Conjecture 3, in particular, for bipartite multigraphs, subcubic multigraphs, and planar graphs of large enough maximum degree For these classes we show that Conjecture 3 holds even when the precoloured set is allowed to be a distance-1 matching. One of our motivations for the formulation and study of Conjecture 3 comes from the close connections with vertex-precolouring and with the LCC
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