Twin-width of random graphs

TL;DR

This paper studies the twin-width of Erdős-Rényi random graphs, revealing a phase transition at p* and providing precise asymptotic estimates for different regimes of p, including dense and sparse graphs.

Contribution

It uncovers a surprising phase transition in the twin-width of G(n,p) and provides asymptotic formulas for both dense and sparse regimes, advancing understanding of this graph parameter.

Findings

Twin-width of G(n,p) is approximately 2p(1-p)n for p in [p*, 1-p*].

Twin-width of G(n,1/2) concentrates around n/2 - sqrt(3n log n)/2.

For sparse graphs, twin-width is Θ(n√p) when p ≥ (726 ln n)/n.

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 $0<p<p^*$ 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 $\Theta(n\sqrt{p})$ when $(726\ln n)/n\leq p\leq1/2$.