A precolouring extension of Vizing's theorem

Abstract

Fix a palette of Δ + 1 colors, a graph with maximum degree Δ, and a subset M of the edge set with minimum distance between edges at least 9. If the edges of M are arbitrarily precoloured from , then there is guaranteed to be a proper edge-coloring using only colors from that extends the precolouring on M to the entire graph. This result is a first general precolouring extension form of Vizing's theorem, and it proves a conjecture of Albertson and Moore under a slightly stronger distance requirement.We also show that the condition on the distance can be lowered to 5 when the graph contains no cycle of length 5. K E Y W O R D Sedge-coloring, precolouring extension, Vizing's theorem

Restricted extension of sparse partial edge colorings of hypercubes

“…In particular, in [12] it is proved that if G is subcubic or bipartite and ϕ is an edge precoloring of a matching M in G using ∆(G) + 1 colors, then ϕ can be extended to a proper (∆(G) + 1)-edge coloring of G, where ∆(G) as usual denotes the maximum degree of G; a similar result on avoiding a precolored matching of a general graph is obtained as well. Moreover, in [16] it is proved that if ϕ is an (∆(G) + 1)-edge precoloring of a distance-9 matching in any graph G, then ϕ can be extended to a proper (∆(G)+1)-edge coloring of G; here, by a distance-9-matching we mean a matching M where the distance between any two edges in M is at least 9; the distance between two edges e and e ′ is the number of edges contained in a shortest path between an endpoint of e, and an endpoint of e ′ . A distance-2 matching is usually called an induced matching.…”

“…One of the earlier references explicitly discussing the problem of extending a partial edge coloring is [23]; there a necessary condition for the existence of an extension is given and the authors find a class of graphs where this condition is also sufficient. More recently, questions on extending and avoiding a precolored matching have been studied in [12,16]. In particular, in [12] it is proved that if G is subcubic or bipartite and ϕ is an edge precoloring of a matching M in G using ∆(G) + 1 colors, then ϕ can be extended to a proper (∆(G) + 1)-edge coloring of G, where ∆(G) as usual denotes the maximum degree of G; a similar result on avoiding a precolored matching of a general graph is obtained as well.…”

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.…”

“…Note that the conjecture on distance‐2 matchings in [5] is sharp both with respect to the distance between precolored edges, and in the sense that can in general not be replaced by (for Class 1 graphs), even if any two precolored edges are at arbitrarily large distance from each other [5]. In [5], it is proved that this conjecture holds for, for example, bipartite multigraphs and subcubic multigraphs, and in [10] it is proved that a version of the conjecture with the distance condition increased to 9 holds for general graphs.…”

“…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 using exactly colors.…”