Coloring Graphs with no Even Hole of Length at Least 6: the Triangle-Free Case.

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 Thomasse to prove that graphs with no induced cycle of length divisible by three have bounded chromatic number.