Edge precoloring extension of hypercubes

Abstract

We consider the problem of extending partial edge colorings of hypercubes. In particular, we obtain an analogue of the positive solution to the famous Evans' conjecture on completing partial Latin squares by proving that every proper partial edge coloring of at most d − 1 edges of the d-dimensional hypercube Q d can be extended to a proper d-edge coloring of Q d . Additionally, we characterize which partial edge colorings of Q d with precisely d precolored edges are extendable to proper d-edge colorings of Q d , and consider some related edge precoloring extension problems of hypercubes.Note that the conjecture on distance-2 matchings in [6] is sharp both with respect to the distance between precolored edges, and in the sense that ∆(G) + 1 can in general not be replaced by ∆(G), even if any two precolored edges are at arbitrarily large distance from each other [6]. In [6], it is proved that this conjecture holds for e.g. bipartite multigraphs and subcubic multigraphs, and in [11] it is proved that a version of the conjecture with the distance increased to 9 holds for general graphs.However, for one specific family of graphs, the balanced complete bipartite graphs K n,n , the edge precoloring extension problem was studied far earlier than in the above-mentioned references.Here the extension problem corresponds to asking whether a partial Latin square can be completed to a Latin square. In this form the problem appeared already in 1960, when Evans [7] stated his now classic conjecture that for every positive integer n, if n−1 edges in K n,n have been (properly) colored, then the partial coloring can be extended to a proper n-edge-coloring of K n,n . This conjecture was solved for large n by Häggkvist [15] and later for all n by Smetaniuk [18], and independently by Andersen and Hilton [2]. Moreover, Andersen and Hilton [2] characterized which n × n partial Latin squares with exactly n non-empty cells are extendable.In this paper we consider the edge precoloring extension problem for the family of hypercubes. Although matching extendability and subgraph containment problems have been studied extensively for hypercubes, (see e.g. [19,9,20] 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 [6, 11]) we consider only proper edge colorings of hypercubes Q d using exactly ∆(Q d ) colors.We prove that every proper 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. Moreover, similarly to [2] we also characterize which proper precolorings with exactly d precolored edges are not extendable to proper d-edge colorings of Q d . We also consider the cases when the precolored edges form an induced matching or one or two hypercubes of smaller dimension. The paper is concluded by a conjecture and some examples and...

Restricted Extension of Sparse Partial Edge Colorings of Complete graphs

“…Similar questions have also been investigated for complete graphs by Andersen and Hilton [2]; as is well-known, problems on extending partial edge colorings of complete graphs can be formulated as questions on completing symmetric partial Latin squares. Moreover, quite recently, Casselgren et al [8] proved an analogue of this result for hypercubes.…”

Restricted extension of sparse partial edge colorings of hypercubes

“…The study of problems on extending and avoiding partial edge colorings of hypercubes was recently initiated in the papers [6,7]. In [6] Casselgren et al obtained several analogues for hypercubes of classic results on completing partial Latin squares, such as the famous Evans conjecture. Moreover, questions on extending a "sparse" precoloring of a hypercube subject to the condition that the extension should avoid a given "sparse" list assignment were investigated in [7].…”

“…Extending and avoiding edge colorings simultaneouslyIn[6], it was proved that any partial proper coloring of at most d − 1 edges of Q d is extendable to a proper d-edge coloring of Q d . Moreover, it was proved that any partial proper coloring of at most d edges in Q d is extendable unless it satisfies one of the following conditions: (C1) there is an uncolored edge uv in Q d such that u is incident with edges of r ≤ d distinct colors and v is incident to d − r edges colored with d − r other distinct colors (so uv is adjacent to edges of d distinct colors);…”

Edge Precoloring Extension of Trees

“…The problem of extending partial edge colorings of hypercubes was recently studied by Casselgren, Markstrom and Pham [9]. They obtained an analogue of the positive solution to Evans' conjecture on completing partial Latin squares by proving that every proper partial edge coloring of at most n − 1 edges of the n-dimensional hypercube Q n can be extended to a proper n-edge coloring of Q n .…”

“…More recently, the problem of extending a precoloring of a matching has been considered in [12]. In particular it is conjectured that for every graph G, if ϕ is an edge precoloring of a matching M in G using ∆(G) + 1 colors, and any two edges in M are at distance at least 2 from each other, then ϕ can be extended to a proper (∆(G) + 1)-edge coloring of G. In addition, with motivation from results on completing partial Latin squares, questions on extending partial edge colorings of d-dimensional hypercubes Q d was studied in [9]. In particular, a characterization of which partial edge colorings with at most d precolored edges are extendable to d-edge colorings of Q d is obtained, thereby establishing an analogue for hypercubes of the characterization by Andersen and Hilton [4] of which n × n partial Latin squares with at most n non-empty cells are completable to Latin squares.…”