Pollyanna and Polynomially \c{hi}-Bounded Graph Classes

A hereditary graph class is called polynomially $\chi$-bounded if there exists a polynomial function $f$ such that $\chi(G) \le f(\omega(G))$ for every induced subgraph $G$. A class $\mathcal{C}$ is called Pollyanna if, for every $\chi$-bounded class $\mathcal{F}$, the class $\mathcal{C} \cap \mathcal{F}$ is polynomially $\chi$-bounded. In the paper by Chudnovsky et al., \emph{Reuniting $\chi$-boundedness with polynomial $\chi$-boundedness} (J.\ Combin.\ Theory Ser.\ B 176 (2026), 30--73), the authors posed twelve problems and one conjecture concerning the Pollyanna framework. In this work, we investigate several of these problems by studying the chromatic number of hereditary graph classes defined by forbidden induced subgraphs. We prove three new strong Pollyanna results. In particular, for every $t \ge 2$, every $\{\text{diamond}, \mathrm{hammer}(t)^+\}$-free graph is $t$-strongly Pollyanna. We also show that graph classes obtained by forbidding suitable combinations of bowties and dumbbells are $(2t-2)$-strongly Pollyanna. We show that the class of $\{(2,2)$-bowtie, $P_5$, $(3,3)$-dumbbell$\}$-free graphs is polynomially $\chi$-bounded. We also prove polynomial $\chi$-boundedness for diamond-free graphs in which every edge lies in at least two triangles, under additional forbidden configurations.