Matchings in Hypercubes Extend to Long Cycles

TL;DR

This paper addresses the Ruskey-Savage conjecture on extending matchings in the $n$-dimensional hypercube $Q_n$ to Hamilton cycles, focusing on matchings that span at most $d$ directions. It introduces a subcube decomposition approach $Q_n = Q_{n-d} \square Q_d$ and proves a conditional Hamilton-cycle extension for general $n$ provided a core conjecture holds in dimension $d$, with the condition verified by computer for $d \le 5$. It also develops a framework for Hamilton-path extensions between opposite-parity endpoints, formalizing C-conditions that block such extensions and proving a conditional, verifiable result for $d \le 5$. Together, these results advance understanding of the Ruskey-Savage problem by resolving near-complete cases unconditionally and by offering a practical computational method to validate the central conjecture in small dimensions, with potential implications for Gray-code-like traversals and hypercube routing.

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 $d=5$.

Figures (3)

• Figure 1: The forbidden case for $n=4$. Orange edges are the matching $M$ and the orange vertices are the vertices $u$, $v$ which cannot be together extended to a Hamilton path extending $M$ between $u$ and $v$. Black edges are the remaining usable edges in $Q_4$.
• Figure 2: Illustration of proof of Theorem \ref{['theorem:hamilton-cycle']}. On the left is a Hamilton cycle $C = (c_1, \dots, c_k)$ on $Q_{n-d}$ and on the right the solid lines are edges between subcubes and the dashed lines are the Hamilton paths on the subcubes.
• Figure 3: Necessary conditions for the existence of Hamilton paths in a hypercube, a) condition C1, $u$ and $v$ covered by a half-layer b) condition C2, $M$ contains $u$ and $v$ avoiding almost half-layer in direction $i$ and $u u^j, v v^j \in M, j \neq i$.

Theorems & Definitions (32)

• theorem 1: Fink kreweras1
• theorem 2: Alahmadi et al. alahmadi2015extending, Theorem 4
• theorem 3: Shujia Wang, Fan Wang wang24
• theorem 4
• theorem 5
• theorem 6: Gregor, Novotný, and Škrekovski gregor2017, Theorem 2
• theorem 7
• theorem 8
• proof : Proof of Theorem \ref{['theorem:hamilton-cycle']}
• proof : Proof of Theorem \ref{['theorem:cycle-dim-5']}