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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. Thisextends a quasi-polynomial bound from Pilipczuk and Soko{\l}owski andgeneralizes a result for bounded clique-width graphs by Bonamy and Pilipczuk.The proof uses the main ideas of the quasi-polynomial approach, with adifferent treatment of the decomposition tree. In particular, we identify twotypes of extensions of a class of graphs: the delayed-extension (whichpreserves polynomial $\chi$-boundedness) and the right-extension (whichpreserves polynomial $\chi$-boundedness under bounded twin-width condition).Our main result is that every bounded twin-width graph is a delayed extensionof simpler classes of graphs, each expressed as a bounded union of rightextensions of lower twin-width graphs.

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