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arXiv · 2303.11231

Bounded twin-width graphs are polynomially $\chi$-bounded

Romain Bourneuf·Stéphan Thomassé
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Abstract

We show that every graph with twin-width $t$ has chromatic number $O(\omega ^{k_t})$ for some integer $k_t$, where $\omega$ denotes the clique number. This extends a quasi-polynomial bound from Pilipczuk and Soko{\l}owski and generalizes a result for bounded clique-width graphs by Bonamy and Pilipczuk. The proof uses the main ideas of the quasi-polynomial approach, with a different treatment of the decomposition tree. In particular, we identify two types of extensions of a class of graphs: the delayed-extension (which preserves polynomial $\chi$-boundedness) and the right-extension (which preserves polynomial $\chi$-boundedness under bounded twin-width condition). Our main result is that every bounded twin-width graph is a delayed extension of simpler classes of graphs, each expressed as a bounded union of right extensions of lower twin-width graphs.

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Romain Bourneuf, Stéphan Thomassé. 2023-03-20. Bounded twin-width graphs are polynomially $\chi$-bounded. https://doi.org/10.19086/aic.2025.2

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