On the analogue of Esperet's conjecture: Characterizing hereditary classes

In the paper [J. Graph Theory (2023) 102:458-471, the Esperet's conjecture has been posed: Every $\chi$-bounded hereditary class is poly-$\chi$-bounded]. This conjecture was first posed in [Habilitation Thesis, Universit\'e Grenoble Alpes, 24, 2017]. This is adapted from the Gy\'arf\'as--Sumner's conjecture which has been asserted in [The Theory and Applications of Graphs, (G. Chartrand, ed.), John Wiley & Sons, New York, 1981, pp. 557-576]. Although the Esperet's conjecture is false in general, in this study we consider an analogue of Esperet's conjecture as follows: Let $C$ be a hereditary class of graphs, and $d \ge 1$. Suppose that there is a function $f$ such that $\chi(G) \le f(\tau_d(G))$ for each $G \in C$. Can we always choose $f$ to be a polynomial? We investigate this conjecture by focusing on specific classes of graphs. This work identifies hereditary graph classes that do not contain specific induced subdivisions of claws and confirms that they adhere to the stated conjecture.