Connecting hypercube 1-factors
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915455733596160 |
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| author | Hollom, Lawrence Shaw, Benedict Randall |
| author_facet | Hollom, Lawrence Shaw, Benedict Randall |
| contents | 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 $Ω(\log d)$ is such that $\bigcup \mathcal{N}$ is connected. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_15698 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Connecting hypercube 1-factors Hollom, Lawrence Shaw, Benedict Randall Combinatorics 05C70 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 $Ω(\log d)$ is such that $\bigcup \mathcal{N}$ is connected. |
| title | Connecting hypercube 1-factors |
| topic | Combinatorics 05C70 |
| url | https://arxiv.org/abs/2508.15698 |