A Note on Chromatic Number and Induced Odd Cycles

An odd hole is an induced odd cycle of length at least 5. Scott and Seymour confirmed a conjecture of Gyárfás and proved that if a graph G has no odd holes then χ(G) 6 22 ω(G)+2 . Chudnovsky, Robertson, Seymour and Thomas showed that if G has neither K4 nor odd holes then χ(G) 6 4. In this note, we show that if a graph G has neither triangles nor quadrilaterals, and has no odd holes of length at least 7, then χ(G) 6 4 and χ(G) 6 3 if G has radius at most 3, and for each vertex u of G, the set of vertices of the same distance to u induces a bipartite subgraph. This answers some questions in [17].