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arXiv · 2401.01769

Matchings in hypercubes extend to long cycles

Jiří Fink·Torsten Mütze
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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.

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Jiří Fink, Torsten Mütze. 2025-04-16. Matchings in hypercubes extend to long cycles. https://arxiv.org/abs/2401.01769

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