Connecting hypercube 1-factors

TL;DR

This paper proves that in the hypercube graph, there exists a 1-factorisation where unions of logarithmically many perfect matchings are connected, improving previous bounds and demonstrating a probabilistic construction.

Contribution

It introduces a random construction showing that unions of logarithmically many 1-factors in the hypercube are connected, significantly improving prior bounds.

Findings

Existence of a 1-factorisation with unions of O(log d) matchings being connected

Improved upper bound from previous loor(d/2)loor to O(log d)

Probabilistic method used for the construction

Abstract

A 1-factorisation of a regular graph $G$ is a partition of its edge set $E(G)$ into perfect matchings of $G$. Behague asked for the minimal $r=r(d)$ such that some $1$-factorisation of the $d$-dimensional hypercube $Q_d$ has the property that the union of any $r$ of its 1-factors is connected. Previous work by Laufer on perfect $1$-factorisations implied that $r$ is at least three, and Behague gave a construction with $r=\big\lceil\frac{d}{2}\big\rceil+1$. We improve this upper bound, giving a random construction with $r=O(\log d)$. In other words, we prove the existence of a 1-factorisation $\mathcal{M} = \{M_1,\dotsc,M_d\}$ of the hypercube $Q_d$ such that every $\mathcal{N}\subseteq \mathcal{M}$ of size $\Omega(\log d)$ is such that $\bigcup \mathcal{N}$ is connected.