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arXiv · 2310.11167

Reuniting $\chi$-boundedness with polynomial $\chi$-boundedness

Maria Chudnovsky·Linda Cook·James Davies·Sang-il Oum
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Abstract

A class $\mathcal{F}$ of graphs is $\chi$-bounded if there is a function $f$ such that $\chi(H)\le f(\omega(H))$ for all induced subgraphs $H$ of a graph in $\mathcal{F}$. If $f$ can be chosen to be a polynomial, we say that $\mathcal{F}$ is polynomially $\chi$-bounded. Esperet proposed a conjecture that every $\chi$-bounded class of graphs is polynomially $\chi$-bounded. This conjecture has been disproved; it has been shown that there are classes of graphs that are $\chi$-bounded but not polynomially $\chi$-bounded. Nevertheless, inspired by Esperet's conjecture, we introduce Pollyanna classes of graphs. A class $\mathcal{C}$ of graphs is Pollyanna if $\mathcal{C}\cap \mathcal{F}$ is polynomially $\chi$-bounded for every $\chi$-bounded class $\mathcal{F}$ of graphs. We prove that several classes of graphs are Pollyanna and also present some proper classes of graphs that are not Pollyanna.

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Maria Chudnovsky, Linda Cook, James Davies, Sang-il Oum. 2023-10-17. Reuniting $\chi$-boundedness with polynomial $\chi$-boundedness. https://doi.org/10.1016/j.jctb.2025.08.002

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