Coloring graphs with no even hole $\geq 6$: the triangle-free case

Abstract: In this paper, we prove that the class of graphs with no triangle and no induced cycle of even length at least 6 has bounded chromatic number. It is well-known that even-hole-free graphs are $\chi$-bounded but we allow here the existence of $C_4$. The proof relies on the concept of Parity Changing Path, an adaptation of Trinity Changing Path which was recently introduced by Bonamy, Charbit and Thomass\'e to prove that graphs with no induced cycle of length divisible by three have bounded chromatic number.