2018
DOI: 10.37236/6303
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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 c… Show more

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Cited by 15 publications
(36 citation statements)
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“…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).…”
Section: Introductionmentioning
confidence: 83%
See 3 more Smart Citations
“…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).…”
Section: Introductionmentioning
confidence: 83%
“…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).…”
Section: Introductionmentioning
confidence: 88%
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“…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.…”
Section: Introductionmentioning
confidence: 99%