Stability of large cuts in random graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917595718877184 |
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| author | Hoshen, Ilay Samotij, Wojciech Zhukovskii, Maksim |
| author_facet | Hoshen, Ilay Samotij, Wojciech Zhukovskii, Maksim |
| contents | We prove that the family of largest cuts in the binomial random graph exhibits the following stability property: If $1/n \ll p = 1-Ω(1)$, then, with high probability, there is a set of $n - o(n)$ vertices that is partitioned in the same manner by all maximum cuts of $G_{n,p}$. Moreover, the analogous statement remains true when one replaces maximum cuts with nearly-maximum cuts.
We then demonstrate how one can use this statement as a tool for showing that certain properties of $G_{n,p}$ that hold in a fixed balanced cut hold simultaneously in all maximum cuts. We provide two example applications of this tool. First, we prove that maximum cuts in $G_{n,p}$ typically partition the neighbourhood of every vertex into nearly equal parts; this resolves a conjecture of DeMarco and Kahn for all but a narrow range of densities $p$. Second, for all edge-critical, nonbipartite, and strictly 2-balanced graphs $H$, we prove a lower bound on the threshold density $p$ above which every largest $H$-free subgraph of $G_{n,p}$ is $(χ(H)-1)$-partite. Our lower bound exactly matches the upper bound on this threshold recently obtained by the first two authors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_14620 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stability of large cuts in random graphs Hoshen, Ilay Samotij, Wojciech Zhukovskii, Maksim Combinatorics Probability We prove that the family of largest cuts in the binomial random graph exhibits the following stability property: If $1/n \ll p = 1-Ω(1)$, then, with high probability, there is a set of $n - o(n)$ vertices that is partitioned in the same manner by all maximum cuts of $G_{n,p}$. Moreover, the analogous statement remains true when one replaces maximum cuts with nearly-maximum cuts. We then demonstrate how one can use this statement as a tool for showing that certain properties of $G_{n,p}$ that hold in a fixed balanced cut hold simultaneously in all maximum cuts. We provide two example applications of this tool. First, we prove that maximum cuts in $G_{n,p}$ typically partition the neighbourhood of every vertex into nearly equal parts; this resolves a conjecture of DeMarco and Kahn for all but a narrow range of densities $p$. Second, for all edge-critical, nonbipartite, and strictly 2-balanced graphs $H$, we prove a lower bound on the threshold density $p$ above which every largest $H$-free subgraph of $G_{n,p}$ is $(χ(H)-1)$-partite. Our lower bound exactly matches the upper bound on this threshold recently obtained by the first two authors. |
| title | Stability of large cuts in random graphs |
| topic | Combinatorics Probability |
| url | https://arxiv.org/abs/2402.14620 |