[Paper Review] Graphs of bounded cliquewidth are polynomially $χ$-bounded

Marthe Bonamy, Michał Pilipczuk|arXiv (Cornell University)|Oct 1, 2019

Advanced Graph Theory Research15 references4 citations

TL;DR

This paper proves that graphs of bounded cliquewidth are polynomially χ-bounded, meaning their chromatic number is bounded by a polynomial function of their clique number. The authors establish a general framework showing that if a hereditary graph class is polynomially χ-bounded, then so is the class of graphs decomposable into pieces from that class along cuts of bounded rank, which directly implies polynomial χ-boundedness for all cliquewidth-k graphs.

ABSTRACT

We prove that if $\mathcal{C}$ is a hereditary class of graphs that is polynomially $χ$-bounded, then the class of graphs that admit decompositions into pieces belonging to $\mathcal{C}$ along cuts of bounded rank is also polynomially $χ$-bounded. In particular, this implies that for every positive integer $k$, the class of graphs of cliquewidth at most $k$ is polynomially $χ$-bounded.

Motivation & Objective

• To establish polynomial χ-boundedness for graphs of bounded cliquewidth, a long-standing open problem in structural graph theory.
• To extend the known χ-boundedness of hereditary graph classes to more complex graph classes built via controlled decompositions.
• To provide a general method that preserves polynomial χ-boundedness under certain graph decomposition operations.
• To resolve the asymptotic tightness of χ-bounding functions for cliquewidth-bounded classes, showing polynomial bounds are optimal in this context.

Proposed method

• The authors introduce a general closure property: if a hereditary graph class C is polynomially χ-bounded, then the class of graphs admitting C-governed decompositions along cuts of bounded rank is also polynomially χ-bounded.
• They define 'diversity' of a cut as the number of distinct neighborhoods between the two parts, and use this to control the complexity of decompositions.
• The proof leverages structural decomposition theorems and inductive coloring arguments on the decomposition tree, ensuring that chromatic number grows polynomially with clique number.
• The method generalizes earlier results by Dvořák and Král’ on χ-boundedness, upgrading them to polynomial bounds via a refined analysis of coloring propagation through decompositions.
• The framework is applied to show that graphs of cliquewidth at most k are polynomially χ-bounded for every fixed k.
• The approach relies on the equivalence between bounded cliquewidth and bounded rankwidth, allowing the use of rank-based decomposition techniques.

Experimental results

Research questions

• RQ1 Can the χ-boundedness of graphs of bounded cliquewidth be strengthened to polynomial χ-boundedness?
• RQ2 Does the closure property of polynomial χ-boundedness under C-governed decompositions along bounded-rank cuts hold for all hereditary classes?
• RQ3 Is the polynomial dependence of the χ-bounding function on the clique number tight for cliquewidth-k graphs?
• RQ4 Can the framework be used to derive polynomial χ-boundedness for other graph classes defined by vertex-minor exclusions?
• RQ5 What is the minimal degree of the polynomial χ-bounding function for graphs of cliquewidth k, and does it grow with k?

Key findings

• For every fixed k ∈ ℕ, the class of graphs of cliquewidth at most k is polynomially χ-bounded, resolving an open problem in χ-boundedness theory.
• The degree of the polynomial χ-bounding function for cliquewidth-k graphs grows with k, and this growth is unavoidable, as shown in Lemma 5.2.
• The general framework proves that if a hereditary class C is polynomially χ-bounded, then so is the class of graphs decomposable into C-parts along cuts of bounded rank.
• The result implies that W₅-vertex-minor-free graphs are polynomially χ-bounded, extending earlier results that only established χ-boundedness.
• The framework unifies and strengthens prior results, including those by Dvořák and Král’ on χ-boundedness and Kim et al. on polynomial χ-boundedness for Cₗ-vertex-minor-free graphs.
• The work supports the conjecture that all H-vertex-minor-free graph classes are polynomially χ-bounded, particularly when H is a circle graph, as shown by recent results from Geelen et al.