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

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.