Twin-width of random graphs

Abstract

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 $(726\ln n)/n\leq p\leq1/2$.

Figures (3)

• Figure 1: Description of partitions $\mathcal{B}^{n,a}_i$ of $[n]$, where $i \in \{1, a+1, a+b+1\}$ and $b := n-2a$. Each gray region represents a part of size at least two in $\mathcal{B}_i$. A part $y_1$ of size $3$ appears first in $\mathcal{B}_{a+1}$, and a part $z_1$ of size $4$ appears first in $\mathcal{B}_{a+b+1}$.
• Figure 2: The left illustrates the partition of $Z$ into vertex sets of size $3$ except one part, and the right is the bipartite trigraph $H$ on bipartition $\{T_1,\ldots,T_k\}\cup\Pi$.
• Figure 3: The red lines are graphs of $y=\alpha(x)$ and the blue lines are graphs of $y=\beta(x)$.

Theorems & Definitions (59)

• Theorem 1.1: Ahn, Hendrey, Kim, and Oum AHKO2021
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Proposition 2.1: See Diestel2018
• Lemma 2.3: Bonnet, Kim, Reinald, and Thomassé twin-width6
• Lemma 2.4
• proof
• Lemma 2.5: Markov's inequality
• Lemma 2.6: Chebyshev's inequaity