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.

翻译：我们证明，每个具有孪生宽度$t$的图，其色数满足$O(\omega ^{k_t})$，其中$k_t$为某个整数，$\omega$表示团数。这扩展了Pilipczuk和Soko{\l}owski的拟多项式界，并推广了Bonamy和Pilipczuk关于有界团宽图的结论。证明采用了拟多项式方法的主要思想，但对分解树做了不同处理。特别地，我们识别了图类的两种扩展形式：延迟扩展（保持多项式χ-有界性）和右扩展（在孪生宽度有界条件下保持多项式χ-有界性）。我们的主要结论是：每个有界孪生宽度图都是更简单图类的延迟扩展，每个简单图类可表示为低孪生宽度图的右扩展的有界并集。