On the chromatic number of graphs of odd girth without longer odd holes

Abstract

An odd hole is an induced odd cycle of length at least five. Let l≥2 be an integer, and let Gl denote the family of graphs which have girth 2l+1 and have no holes of odd length at least 2l+3. Chudnovsky and Seymour proved that every graph in G2 is three-colorable. Following the idea of Chudnovsky and Seymour, Wu, Xu and Xu proved that every graph in G3 is three-colorable. In 2022, Wu, Xu and Xu conjectured that every graph G∈⋃l≥2Gl is three-colorable. In this paper, we prove that every graph G∈Gl with radius at most l+3 is three-colorable.