Abstract
AbstractWe 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 edges of the ‐dimensional hypercube can be extended to a proper ‐edge coloring of . Additionally, we characterize which partial edge colorings of with precisely precolored edges are extendable to proper ‐edge colorings of .
Highlights
An edge precoloring of a graph G is a proper edge coloring of some subset E′ ⊆ E (G); a t‐edge precoloring is such a coloring with t colors
We prove that every edge precoloring of the d‐dimensional hypercube Qd with at most d − 1 precolored edges is extendable to a d‐edge coloring of Qd, thereby establishing an analogue of the positive resolution of Evans' conjecture
There is a proper edge coloring of J2 = Qd − E (J1) using Δ(J2) colors, and which agrees with the restriction of φ to J2, because J2 is a collection of disjoint one‐ or two‐dimensional hypercubes, where every component contains at most one precolored edge
Summary
An edge precoloring (or partial edge coloring) of a graph G is a proper edge coloring of some subset E′ ⊆ E (G); a t‐edge precoloring is such a coloring with t colors. If φ is an edge precoloring of Qd where all precolored edges lie in an induced matching, φ is extendable to a proper d‐edge coloring.
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