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 ∆. 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.

List-edge-colouring planar graphs with precoloured edges

“…The case d = t = 1 of Theorem 5 was previously established by Edwards, Girão, van den Heuvel, Kang, Sereni and the third author [5], with the slightly stronger assumption of ∆ ≥ 19. (Note that the restriction of our proof for Theorem 2 to this case provides a somewhat new proof; both arguments use global discharging, but we discharge in a different way).…”

“…As Edwards et al [5] observed, extending an edge-colouring is closely related to list-edge-colouring. An edge list assignment on a graph G is a function L that assigns to each edge e ∈ E(G) a list of colours L(e).…”

“…Edwards et al [5] applied Theorem 6 to obtain a precolouring extension result for bipartite graphs (Theorem 15 of [5]), which we will use as part of our proof. While the result as stated in [5] only applies to classical edge-precolouring, a list-edge-colouring version can be obtained using essentially the same proof:…”

“…After the seminal work of Marcotte and Seymour [9], the vertex-version of the precolouring extension problem received much more attention than Question 1. Edwards et al [5] re-initiated this study in their paper, with planar graphs being only one of the many families they considered. The main concern in [5] however is when H is a matching, and in order to guarantee extensions they often impose distance conditions on the edges in the precoloured matching.…”

Edge precoloring extension of hypercubes

“…Although matching extendability and subgraph containment problems have been studied extensively for hypercubes (see, eg, [8,11,18,19] and references therein), the edge precoloring extension problem for hypercubes seems to be a hitherto quite unexplored line of research. As in the setting of completing partial Latin squares (and unlike the papers [5,10]) we consider only proper edge colorings of hypercubes Q d using exactly Q Δ( ) d colors. We prove that every edge precoloring of the d-dimensional hypercube Q d with at most d − 1 precolored edges is extendable to a d-edge coloring of Q d , thereby establishing an analogue of the positive resolution of Evans' conjecture.…”