A class $\mathcal{F}$ of graphs is $χ$-bounded if there is a function $f$ such that $χ(H)\le f(ω(H))$ 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 $χ$-bounded. Esperet proposed a conjecture that every $χ$-bounded class of graphs is polynomially $χ$-bounded. This conjecture has been disproved; it has been shown that there are classes of graphs that are $χ$-bounded but not polynomially $χ$-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 $χ$-bounded for every $χ$-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.

翻译：若存在函数f，使得图类$\mathcal{F}$中任意图的导出子图H均满足$χ(H)\le f(ω(H))$，则称$\mathcal{F}$是χ-有界的。若f可选取为多项式函数，则称$\mathcal{F}$是多项式χ-有界的。Esperet曾提出猜想：每个χ-有界图类都是多项式χ-有界的。该猜想已被证伪；研究证明存在χ-有界但非多项式χ-有界的图类。尽管如此，受Esperet猜想的启发，我们引入了Pollyanna图类的概念。若图类$\mathcal{C}$与任意χ-有界图类$\mathcal{F}$的交集$\mathcal{C}\cap \mathcal{F}$都是多项式χ-有界的，则称$\mathcal{C}$为Pollyanna图类。我们证明了若干图类具有Pollyanna性质，同时也给出了非Pollyanna图类的具体实例。