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，若图G的孪生宽度为t，则其色数χ满足χ ≤ O(ω^{k_t})，其中ω为团数，k_t为某个整数。该结果将Pilipczuk和Sokołowski的拟多项式界进行了推广，并推广了Bonamy和Pilipczuk关于有界团宽度图的结果。证明采用拟多项式方法的核心思想，但对分解树进行了不同处理。特别地，我们识别出图类的两类扩展：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，其中每个子类都可表示为较低孪生宽度图右扩展的有界并集。