Matchings in hypercubes extend to long cycles

TL;DR

This work addresses the Ruskey-Savage conjecture by proving that every matching of the $d$-dimensional hypercube $Q_d$ ($d\ge2$) can be extended to a cycle that visits at least $2/3$ of the vertices. The authors develop an induction framework on dimension, anchored by an auxiliary theorem that extends a matching while avoiding a forbidden vertex $z$, and they introduce structural tools based on half-layers and quad-layers to manage the combinatorics. Key technical components include forward and reverse implications for the forbidden-vertex theorem, detailed handling of $z$-dangerous layers, and a computational base case at $d=5$ to seal the induction. The results generalize to the complete graph on the hypercube’s vertex set, $K(Q_d)$, yielding long cycles and, via Hamilton-laceability connections, consequences for Hamilton paths with endpoints of opposite parity. The paper also discusses broader restricted Gray codes and provides implementation details for the small-dimension base case, offering a step toward the full Ruskey-Savage conjecture and raising questions about algorithmic construction and cycle-factor vs Hamilton cycle extendability.

Abstract

The $d$-dimensional hypercube graph $Q_d$ has as vertices all subsets of $\{1,\ldots,d\}$, and an edge between any two sets that differ in a single element. The Ruskey-Savage conjecture asserts that every matching of $Q_d$, $d\ge 2$, can be extended to a Hamilton cycle, i.e., to a cycle that visits every vertex exactly once. We prove that every matching of $Q_d$, $d\ge 2$, can be extended to a cycle that visits at least a $2/3$-fraction of all vertices.

Figures (9)

• Figure 1: Cycles and matchings in the hypercube $Q_4$: (a) a Hamilton cycle, where the vertex labels omit curly brackets and commas for conciseness; (b) a perfect matching $M$ and an extension of $M$ to a Hamilton cycle; (c) a maximal, but not perfect matching $M'$; (d) an extension of $M'$ to a cycle factor with two cycles; (e) an extension of $M'$ to a (non-Hamilton) cycle; (f) an extension of $M'$ to a Hamilton cycle, obtained by joining the two cycles in (d).
• Figure 2: Illustration of Hamilton cycles in $Q_d$, $d=4$, subject to various constraints: (a) a set $E$ of $2d-3=5$ edges and a Hamilton cycle extending it; (b) a set $F$ of $2d-5=3$ edges and a Hamilton cycle avoiding it; (c) a perfect matching $M$ and a Hamilton cycle avoiding it; (d) a set $F$ of $\binom{d}{2}-2=4$ vertices and a cycle of length $2^d-2|F|=8$ avoiding it.
• Figure 3: The red edges in $Q_4$ form a (a) half-layer, (b) near half-layer with extension vertices $v_1$ and $v_2$, or (c) matching that includes the covered near half-layer $\{e_1,e_2,e_3\}$. The subset of red edges incident with a vertex in $Q^i_1$ form a (a) quad-layer, (b) near quad-layer, or (c) matching that includes the covered near quad-layer $\{e_3\}$. Every vertex $x$ in (a) and (b) colored blue is one for which the (near) quad-layer contained in $Q^i_1$ is $x$-dangerous. The dashed vertical line separates $Q^i_0$ and $Q^i_1$.
• Figure 4: Illustration of the definition of the matching $N$ obtained by modifying $M$, depending on whether $u \notin V(M)$, $u^M \in V(Q^i_0)$ or $u^M \in V(Q^i_1)$ (three columns) and similarly for $u^i$ (three rows). The dotted black line is the non-edge $uu^i\notin M$. The red edges from $M$ are removed, and the green edge is added to $N$. Blue vertices have to be avoided by the cycles $C_0$ and $C_1$ that extend $N^i_0\cup P_0$ and $N^i_1\cup P_1$, respectively.
• Figure 5: Illustration of Case aii in the proof of Theorem \ref{['thm:forbidden']}.

Theorems & Definitions (39)

• Conjecture 1: MR1201997
• Theorem 2: MR2354719
• Theorem 3: MR3891930
• Theorem 4: MR3936192
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9: MR3830138
• Theorem 10: MR3830138