Graphs of bounded twin-width are quasi-polynomially $\chi$-bounded

TL;DR

This paper proves that graphs with bounded twin-width have chromatic numbers that grow quasi-polynomially with their clique number, advancing understanding of their coloring properties.

Contribution

It establishes that classes of graphs with bounded twin-width are quasi-polynomially chi-bounded, a step towards proving they are polynomially chi-bounded.

Findings

Graphs with twin-width at most t have chromatic number bounded by a quasi-polynomial function of their clique number.

This result advances the understanding of coloring properties of graphs with bounded twin-width.

The work makes progress towards resolving whether such graphs are polynomially chi-bounded.

Abstract

We prove that for every $t\in \mathbb{N}$ there is a constant $\gamma_t$ such that every graph with twin-width at most $t$ and clique number $\omega$ has chromatic number bounded by $2^{\gamma_t \log^{4t+3} \omega}$. In other words, we prove that graph classes of bounded twin-width are quasi-polynomially $\chi$-bounded. This provides a significant step towards resolving the question of Bonnet et al. [ICALP 2021] about whether they are polynomially $\chi$-bounded.