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Bounded twin-width graphs are polynomially $χ$-bounded

Romain Bourneuf, Stéphan Thomassé

TL;DR

This work proves that graphs of bounded twin-width have chromatic number polynomially bounded in terms of their clique number, i.e., for fixed twin-width $t$, there exists $k_t$ with $\chi(G) = O(\omega(G)^{k_t})$. The authors introduce two core extension mechanisms—delayed extension and right extension—and show that delayed extensions preserve polynomial $\chi$-boundedness and right extensions preserve (at least) $\chi$-boundedness, with polynomial bounds under bounded twin-width. Central to the approach is a delayed decomposition tree that encodes graphs via simpler components $g(x)$, together with the notion of $d$-almost mixed minors and the Marcus–Tardos theorem to bound mixed zones; this yields that $d$-almost mixed free graphs are polynomially $\chi$-bounded, which cascades to the main result for bounded twin-width graphs. The paper also develops streamlined proofs for substitution-closure results and introduces mixed extensions, broadening the toolkit for establishing $\chi$-boundedness in various graph-structure settings with implications for both theory and potential algorithmic applications.

Abstract

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.

Bounded twin-width graphs are polynomially $χ$-bounded

TL;DR

This work proves that graphs of bounded twin-width have chromatic number polynomially bounded in terms of their clique number, i.e., for fixed twin-width , there exists with . The authors introduce two core extension mechanisms—delayed extension and right extension—and show that delayed extensions preserve polynomial -boundedness and right extensions preserve (at least) -boundedness, with polynomial bounds under bounded twin-width. Central to the approach is a delayed decomposition tree that encodes graphs via simpler components , together with the notion of -almost mixed minors and the Marcus–Tardos theorem to bound mixed zones; this yields that -almost mixed free graphs are polynomially -bounded, which cascades to the main result for bounded twin-width graphs. The paper also develops streamlined proofs for substitution-closure results and introduces mixed extensions, broadening the toolkit for establishing -boundedness in various graph-structure settings with implications for both theory and potential algorithmic applications.

Abstract

We show that every graph with twin-width has chromatic number for some integer , where 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 -boundedness) and the right-extension (which preserves polynomial -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.
Paper Structure (11 sections, 20 theorems, 3 equations, 2 figures)

This paper contains 11 sections, 20 theorems, 3 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Polynomial $\chi$-bounded classes
  3. Twin-width
  4. Overview of the proof
  5. Future directions
  6. First operation: Delayed extension
  7. Second operation: Right extension
  8. Twin-width and almost-mixed minors
  9. Polynomial $\chi$-boundedness of bounded twin-width graphs
  10. More on the $\chi$-boundedness of substitution-closure
  11. Mixed extensions

Key Result

Theorem 1.1

For every $t \in \mathbb{N}$ there is a constant $\gamma_t$ such that every graph with twin-width at most $t$ and clique number $\omega$ has chromatic number bounded by $2^{\gamma_t \log^{4t+3}\omega}$.

Figures (2)

  • Figure 1: A delayed decomposition tree of $C_5$ and its corresponding realization. The edges of every $g(x)$ are drawn in the same color as $x$. Note that all $g(x)$ are cographs.
  • Figure 2: The RMP for $S_{5, 2}$. Here, an edge means that we have all edges from the stable set on the left to the vertex on the right. Observe that every transversal is a forest, however we can form every graph on 4 vertices as a transversal minor.

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Corollary 2.3
  • proof
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • ...and 28 more