Sharp Threshold for Cliques in Random 0/1 Polytope Graphs
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Babecki, Catherine Elling, Tycho Ferber, Asaf |
| author_facet | Babecki, Catherine Elling, Tycho Ferber, Asaf |
| contents | We study graph-theoretic properties of random $0/1$ polytopes. Specifically, let $Q_p^n \subseteq \{0,1\}^n$ be a random subset where each point is included independently with probability $p$, and consider the graph $G_p$ of the polytope conv$(Q_p^n)$. We provide a short and combinatorial proof that $p = 2^{-n/2}$ is a threshold for the edge density of $G_p$, a result originally due to Kaibel and Remshagen. We next resolve an open question from their paper by showing that for $p \leq 2^{-n/2 - o(1)}$, $G_p$ exhibits strong edge expansion. In particular, we prove that, with high probability, every vertex has degree $(1 - o(1))|Q_p^n|$. Lastly, we determine the threshold for $G_p$ being a clique, strengthening a result of Bondarenko and Brodskiy. We show that with high probability, if $p \geq 2^{-δn + o(1)}$, then $G_p$ is not a clique, and if $ p \leq 2^{-δn - o(1)}$, then $G_p$ is a clique, where $δ\approx 0.8295$. Our approach combines a combinatorial characterization of edges in graphs arising from polytopes with the Kim-Vu polynomial concentration inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_03212 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp Threshold for Cliques in Random 0/1 Polytope Graphs Babecki, Catherine Elling, Tycho Ferber, Asaf Combinatorics Discrete Mathematics 52B12, 05C75, 52B05, 05D40, 60C05 We study graph-theoretic properties of random $0/1$ polytopes. Specifically, let $Q_p^n \subseteq \{0,1\}^n$ be a random subset where each point is included independently with probability $p$, and consider the graph $G_p$ of the polytope conv$(Q_p^n)$. We provide a short and combinatorial proof that $p = 2^{-n/2}$ is a threshold for the edge density of $G_p$, a result originally due to Kaibel and Remshagen. We next resolve an open question from their paper by showing that for $p \leq 2^{-n/2 - o(1)}$, $G_p$ exhibits strong edge expansion. In particular, we prove that, with high probability, every vertex has degree $(1 - o(1))|Q_p^n|$. Lastly, we determine the threshold for $G_p$ being a clique, strengthening a result of Bondarenko and Brodskiy. We show that with high probability, if $p \geq 2^{-δn + o(1)}$, then $G_p$ is not a clique, and if $ p \leq 2^{-δn - o(1)}$, then $G_p$ is a clique, where $δ\approx 0.8295$. Our approach combines a combinatorial characterization of edges in graphs arising from polytopes with the Kim-Vu polynomial concentration inequality. |
| title | Sharp Threshold for Cliques in Random 0/1 Polytope Graphs |
| topic | Combinatorics Discrete Mathematics 52B12, 05C75, 52B05, 05D40, 60C05 |
| url | https://arxiv.org/abs/2507.03212 |