Avoiding and extending partial edge colorings of hypercubes

TL;DR

This paper investigates conditions under which partial edge colorings of hypercubes can be extended to full proper colorings that either agree or disagree with the partial coloring, providing new bounds and characterizations.

Contribution

It establishes new sufficient conditions for avoiding partial colorings of hypercubes and characterizes extension possibilities for mixed partial colorings involving multiple edges.

Findings

Avoidability when each color appears on at most d/8 edges under mild conditions

Extension results for color classes that are induced matchings when d is divisible by 3

Complete characterization of extension possibilities for configurations with partial colorings of d-k and k edges

Abstract

We consider the problem of extending and avoiding partial edge colorings of hypercubes; that is, given a partial edge coloring $\varphi$ of the $d$-dimensional hypercube $Q_d$, we are interested in whether there is a proper $d$-edge coloring of $Q_d$ that agrees with the coloring $\varphi$ on every edge that is colored under $\varphi$; or, similarly, if there is a proper $d$-edge coloring that disagrees with $\varphi$ on every edge that is colored under $\varphi$. In particular, we prove that for any $d\geq 1$, if $\varphi$ is a partial $d$-edge coloring of $Q_d$, then $\varphi$ is avoidable if every color appears on at most $d/8$ edges and the coloring satisfies a relatively mild structural condition, or $\varphi$ is proper and every color appears on at most $d-2$ edges. We also show that the same conclusion holds if $d$ is divisible by $3$ and every color class of $\varphi$ is an induced matching. Moreover, for all $1 \leq k \leq d$, we characterize for which configurations consisting of a partial coloring $\varphi$ of $d-k$ edges and a partial coloring $\psi$ of $k$ edges, there is an extension of $\varphi$ that avoids $\psi$.