Nearly spanning cycle in the percolated hypercube
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915276264570880 |
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| author | Anastos, Michael Diskin, Sahar Erde, Joshua Kang, Mihyun Krivelevich, Michael Lichev, Lyuben |
| author_facet | Anastos, Michael Diskin, Sahar Erde, Joshua Kang, Mihyun Krivelevich, Michael Lichev, Lyuben |
| contents | Let $Q^d$ be the $d$-dimensional binary hypercube. We form a random subgraph $Q^d_p\subseteq Q^d$ by retaining each edge of $Q^d$ independently with probability $p$. We show that, for every constant $\varepsilon>0$, there exists a constant $C=C(\varepsilon)>0$ such that, if $p\ge C/d$, then with high probability $Q^d_p$ contains a cycle of length at least $(1-\varepsilon)2^d$. This confirms a long-standing folklore conjecture, stated in particular by Condon, Espuny Díaz, Girão, Kühn, and Osthus [Hamiltonicity of random subgraphs of the hypercube, Mem. Amer. Math. Soc. 305 (2024), No. 1534]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_04436 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nearly spanning cycle in the percolated hypercube Anastos, Michael Diskin, Sahar Erde, Joshua Kang, Mihyun Krivelevich, Michael Lichev, Lyuben Combinatorics Probability Let $Q^d$ be the $d$-dimensional binary hypercube. We form a random subgraph $Q^d_p\subseteq Q^d$ by retaining each edge of $Q^d$ independently with probability $p$. We show that, for every constant $\varepsilon>0$, there exists a constant $C=C(\varepsilon)>0$ such that, if $p\ge C/d$, then with high probability $Q^d_p$ contains a cycle of length at least $(1-\varepsilon)2^d$. This confirms a long-standing folklore conjecture, stated in particular by Condon, Espuny Díaz, Girão, Kühn, and Osthus [Hamiltonicity of random subgraphs of the hypercube, Mem. Amer. Math. Soc. 305 (2024), No. 1534]. |
| title | Nearly spanning cycle in the percolated hypercube |
| topic | Combinatorics Probability |
| url | https://arxiv.org/abs/2505.04436 |