Bounds for the Twin-width of Graphs

TL;DR

This paper establishes upper bounds for the twin-width of various classes of graphs, provides lower bounds for specific graph families, and analyzes the asymptotic behavior of twin-width in random graphs.

Contribution

It introduces bounds for twin-width based on graph size and degree, and characterizes twin-width thresholds in random graph models.

Findings

Twin-width of n-vertex graphs is less than (n+√(n ln n)+√n+2 ln n)/2.

Twin-width of m-edge graphs is less than √(3m)+ m^{1/4} √(ln m)/(4·3^{1/4}) + 3m^{1/4}/2.

Conference graphs of order n have twin-width at least (n-1)/2, achieved by Paley graphs.

Abstract

Bonnet, Kim, Thomass\'{e}, and Watrigant (2020) introduced the twin-width of a graph. We show that the twin-width of an $n$-vertex graph is less than $(n+\sqrt{n\ln n}+\sqrt{n}+2\ln n)/2$, and the twin-width of an $m$-edge graph for a positive $m$ is less than $\sqrt{3m}+ m^{1/4} \sqrt{\ln m} / (4\cdot 3^{1/4}) + 3m^{1/4} / 2$. Conference graphs of order $n$ (when such graphs exist) have twin-width at least $(n-1)/2$, and we show that Paley graphs achieve this lower bound. We also show that the twin-width of the Erd\H{o}s-R\'{e}nyi random graph $G(n,p)$ with $1/n\leq p=p(n)\leq 1/2$ is larger than $2p(1-p)n - (2\sqrt{2}+\varepsilon)\sqrt{p(1-p)n\ln n}$ asymptotically almost surely for any positive $\varepsilon$. Lastly, we calculate the twin-width of random graphs $G(n,p)$ with $p\leq c/n$ for a constant $c<1$, determining the thresholds at which the twin-width jumps from $0$ to $1$ and from $1$ to $2$.