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

TL;DR

This paper explores an analogue of Esperet's conjecture, investigating whether certain hereditary graph classes with bounded chromatic number relative to a subdivision parameter can be characterized by polynomial functions, focusing on classes excluding specific induced subdivisions.

Contribution

It identifies hereditary graph classes excluding certain induced subdivisions that satisfy the polynomial bound conjecture, providing new insights into graph coloring properties.

Findings

Hereditary classes excluding specific induced subdivisions adhere to polynomial bounds.

The study confirms the conjecture for classes without certain induced claw subdivisions.

Results contribute to understanding chromatic bounds in hereditary graph classes.

Abstract

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.