Edge precoloring extension of hypercubes

TL;DR

This paper proves that partial edge colorings of hypercubes with up to d-1 precolored edges can be extended to full proper d-edge colorings, generalizing a classical conjecture and characterizing extendability for exactly d precolored edges.

Contribution

It provides a hypercube-specific analogue of Evans' conjecture, establishing extendability conditions for partial edge colorings and characterizing cases with exactly d precolored edges.

Findings

Extension is always possible for up to d-1 precolored edges.

Characterization of extendability when exactly d edges are precolored.

Related extension problems for hypercube edge colorings.

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.