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arXiv · 2212.07880

Twin-width of random graphs

Jungho Ahn·Debsoumya Chakraborti·Kevin Hendrey·Donggyu Kim·Sang-il Oum
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Abstract

We investigate the twin-width of the Erd\H{o}s-R\'enyi 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 $0 p>1-p^*$, 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 $\Theta(n\sqrt{p})$ when $(726\ln n)/n\leq p\leq1/2$.

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Jungho Ahn, Debsoumya Chakraborti, Kevin Hendrey, Donggyu Kim, Sang-il Oum. 2022-12-15. Twin-width of random graphs. https://doi.org/10.1002/rsa.21247

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