Twin-width of random graphs
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866912080352772096 |
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| author | Ahn, Jungho Chakraborti, Debsoumya Hendrey, Kevin Kim, Donggyu Oum, Sang-il |
| author_facet | Ahn, Jungho Chakraborti, Debsoumya Hendrey, Kevin Kim, Donggyu Oum, Sang-il |
| contents | We investigate the twin-width of the Erdős-Rényi random graph $G(n,p)$. We unveil a surprising behavior of this parameter by showing the existence of a constant $p^*\approx 0.4$ such that with high probability, when $p^*\le p\le 1-p^*$, the twin-width is asymptotically $2p(1-p)n$, whereas, when $0<p<p^*$ or $1>p>1-p^*$, the twin-width is significantly higher than $2p(1-p)n$. In addition, we show that the twin-width of $G(n,1/2)$ is concentrated around $n/2 - \sqrt{3n \log n}/2$ within an interval of length $o(\sqrt{n\log n})$. For the sparse random graph, we show that with high probability, the twin-width of $G(n,p)$ is $Θ(n\sqrt{p})$ when $(726\ln n)/n\leq p\leq1/2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_07880 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Twin-width of random graphs Ahn, Jungho Chakraborti, Debsoumya Hendrey, Kevin Kim, Donggyu Oum, Sang-il Combinatorics Discrete Mathematics 05C35 We investigate the twin-width of the Erdős-Rényi random graph $G(n,p)$. We unveil a surprising behavior of this parameter by showing the existence of a constant $p^*\approx 0.4$ such that with high probability, when $p^*\le p\le 1-p^*$, the twin-width is asymptotically $2p(1-p)n$, whereas, when $0<p<p^*$ or $1>p>1-p^*$, the twin-width is significantly higher than $2p(1-p)n$. In addition, we show that the twin-width of $G(n,1/2)$ is concentrated around $n/2 - \sqrt{3n \log n}/2$ within an interval of length $o(\sqrt{n\log n})$. For the sparse random graph, we show that with high probability, the twin-width of $G(n,p)$ is $Θ(n\sqrt{p})$ when $(726\ln n)/n\leq p\leq1/2$. |
| title | Twin-width of random graphs |
| topic | Combinatorics Discrete Mathematics 05C35 |
| url | https://arxiv.org/abs/2212.07880 |