Bounded twin-width graphs are polynomially χ-bounded
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We show that every graph with twin-width t has chromatic number O(ω ^k_t) for some integer k_t, where ω denotes the clique number. This extends a quasi-polynomial bound from Pilipczuk and Sokołowski 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 χ-boundedness) and the right-extension (which preserves polynomial χ-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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