Matchings in Hypercubes Extend to Long Cycles

TL;DR

This paper proves that certain matchings in high-dimensional hypercubes, specifically those involving up to five directions, can be extended into Hamilton cycles or paths, advancing understanding of the Ruskey-Savage conjecture.

Contribution

It extends the Ruskey-Savage conjecture by proving matchings with up to five directions can be extended into Hamilton cycles or paths in hypercubes.

Findings

Matchings with edges spanning up to 5 directions can be extended into Hamilton cycles.

Characterization of when such matchings can form Hamilton paths between specific vertices.

Proofs applicable for arbitrary dimensions up to n, assuming extension properties in lower dimensions.

Abstract

The $n$-dimensional hypercube graph $Q_n$ has as vertices all subsets of $\{1, \ldots, n\}$, and an edge between any two sets that differ in a single element. The Ruskey-Savage conjecture states that every matching of the $n$-dimensional hypercube $Q_n$ can be extended into a Hamilton cycle. We prove that matchings of $Q_n$ containing edges spanning at most $d = 5$ directions can be extended into a Hamilton cycle. We also characterize when these matchings of most $d = 5$ directions can be extended into a Hamilton path between two prescribed vertices. Our proofs work for arbitrary $d$ and $n$ where $d \le n$ assuming some extension properties hold in $Q_d$ which we verified by a computer for $d=5$.