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

TL;DR

This paper proves that triangle-free graphs without even cycles of length at least 6 have bounded chromatic number, extending known results to include graphs with a 4-cycle but no larger even holes.

Contribution

It introduces the concept of Parity Changing Path to establish bounded chromatic number in a new class of graphs, generalizing previous results.

Findings

Graphs with no triangle and no large even holes are chromatically bounded

Parity Changing Path is an effective tool for coloring problems

Extends known results to include graphs with 4-cycles but no larger even holes

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.