Reuniting χ-boundedness with polynomial χ-boundedness

A class F of graphs is χ-bounded if there is a function f such that χ(H)≤f(ω(H)) for all induced subgraphs H of a graph in F. If f can be chosen to be a polynomial, we say that 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 C of graphs is Pollyanna if C∩F is polynomially χ-bounded for every χ-bounded class F of graphs. We prove that several classes of graphs are Pollyanna and also present some proper classes of graphs that are not Pollyanna.