Bounded twin-width graphs are polynomially χ-bounded

2023-03-20

arXiv.org

AI Summary

This paper addresses the chromatic number upper bound problem for graphs of bounded twin-width (twin-width ≤ t). Regarding χ-boundedness, it establishes, for the first time, a **polynomial relationship** between twin-width and chromatic number: χ(G) = O(ω(G)^{k_t}), where ω(G) is the clique number and k_t depends solely on t. Methodologically, the authors introduce two novel decomposition paradigms—“delayed expansion” and “right expansion”—to expose hierarchical structural properties; they then apply structural induction on decomposition trees, recursively model class expansions, and derive tight recurrence-based estimates for chromatic numbers, thereby strictly improving prior quasi-polynomial bounds to polynomial ones. This result unifies and generalizes known χ-boundedness results for classical graph classes such as those of bounded clique-width and bounded rank-width, providing a fundamental chromatic characterization within twin-width theory.

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.

Problem

• Graph Coloring
• Bounded Twin Cover
• Maximum Clique Size

Innovation

• Bounded Twinwidth Graphs
• Coloring Number Limit
• Decomposition Tree Techniques