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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2501.13730 |
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| _version_ | 1866916605566386176 |
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| author | Benjamini, Itai Kalifa, Or Tzalik, Elad |
| author_facet | Benjamini, Itai Kalifa, Or Tzalik, Elad |
| contents | A graph $G$ is $m$-minor-universal if every graph with at most $m$ edges (and no isolated vertices) is a minor of $G$. We prove that the $d$-dimensional hypercube, $Q_d$, is $Ω\left(\frac{2^d}{d}\right)$-minor-universal, and that there exists an absolute constant $C >0$ such that $Q_d$ is not $\frac{C2^d}{\sqrt{d}}$-minor-universal. Similar results are obtained in a more generalized setting, where we bound the size of minors in a product of finite connected graphs. A key component of our proof is the following claim regarding the decomposition of a permutation of a box into simpler, one-dimensional permutations: Let $n_1, \dots, n_d$ be positive integers, and define $X := [n_1] \times \dots \times [n_d]$. We prove that every permutation $σ: X \to X$ can be expressed as $σ= σ_1 \circ \dots \circ σ_{2d-1}$, where each $σ_i$ is a one-dimensional permutation, meaning it fixes all coordinates except possibly one. We discuss future directions and pose open problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_13730 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hypercube minor-universality Benjamini, Itai Kalifa, Or Tzalik, Elad Combinatorics Metric Geometry A graph $G$ is $m$-minor-universal if every graph with at most $m$ edges (and no isolated vertices) is a minor of $G$. We prove that the $d$-dimensional hypercube, $Q_d$, is $Ω\left(\frac{2^d}{d}\right)$-minor-universal, and that there exists an absolute constant $C >0$ such that $Q_d$ is not $\frac{C2^d}{\sqrt{d}}$-minor-universal. Similar results are obtained in a more generalized setting, where we bound the size of minors in a product of finite connected graphs. A key component of our proof is the following claim regarding the decomposition of a permutation of a box into simpler, one-dimensional permutations: Let $n_1, \dots, n_d$ be positive integers, and define $X := [n_1] \times \dots \times [n_d]$. We prove that every permutation $σ: X \to X$ can be expressed as $σ= σ_1 \circ \dots \circ σ_{2d-1}$, where each $σ_i$ is a one-dimensional permutation, meaning it fixes all coordinates except possibly one. We discuss future directions and pose open problems. |
| title | Hypercube minor-universality |
| topic | Combinatorics Metric Geometry |
| url | https://arxiv.org/abs/2501.13730 |