Twin-width of sparse random graphs

TL;DR

This work determines the typical and extremal behavior of the twin-width in sparse graphs, establishing a tight bound $tww(G) \le n^{\frac{d-2}{2d-2}+o(1)}$ for $n$-vertex $d$-regular graphs (with equality for almost all such graphs) and extending the analysis to sparse Erdős--Rényi and regular random graphs via the maximum average degree $\text{mad}(G)$. The authors develop a contraction-based upper-bound framework by mapping $G$ to structured targets through randomized homomorphisms into graphs built from $\Gamma(S,r)$ and $\Pi$, and then bounding the contraction sequences using 2-factor decompositions, Hall’s theorem, and equitable colorings. Complementing these upper bounds, they prove matching lower bounds via counting arguments, showing that in several sparse regimes the typical twin-width grows as $\Theta\big(n^{\frac{d-2}{2d-2}} \cdot \frac{d}{\sqrt{\log n}}\big)$ up to polylog factors, thereby addressing questions about whether twin-width can be as small as $\Theta(\sqrt{e(G)})$ in these regimes. The results illuminate how $\text{mad}(G)$ governs twin-width and provide probabilistic bounds for $G(n,p)$ and $G(n,m)$, with implications for first-order model checking on sparse graph classes.

Abstract

We show that the twin-width of every $n$-vertex $d$-regular graph is at most $n^{\frac{d-2}{2d-2}+o(1)}$ and that almost all $d$-regular graphs attain this bound. More generally, we obtain bounds on the twin-width of sparse Erdős-Renyi and regular random graphs, complementing the bounds in the denser regime due to Ahn, Chakraborti, Hendrey, Kim and Oum.

Theorems & Definitions (35)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4: Upper bounds
• Theorem 1.5: Lower bounds
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3