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.

翻译：如果一个图类 $\mathcal F$ 满足：对于 $\mathcal F$ 中任意图的每个诱导子图 $H$，存在函数 $f$ 使得 $\chi(H)\le f(\omega(H))$，则称此类 $\mathcal F$ 为 $\chi$-有界的。若 $f$ 可选为多项式函数，则称 $\mathcal F$ 为多项式 $\chi$-有界的。Esperet 提出猜想：每个 $\chi$-有界图类都是多项式 $\chi$-有界的。该猜想已被证伪；已证明存在 $\chi$-有界但非多项式 $\chi$-有界的图类。受 Esperet 猜想启发，我们引入 Pollyanna 图类。若图类 $\mathcal C$ 满足：对任意 $\chi$-有界图类 $\mathcal F$，$\mathcal C\cap \mathcal F$ 均为多项式 $\chi$-有界的，则称 $\mathcal C$ 为 Pollyanna 图类。我们证明若干图类为 Pollyanna 图类，同时给出一些非 Pollyanna 的真图类。