ELECTRONIC JOURNAL OF COMBINATORICS
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An odd hole is an induced odd cycle of length at least 5. Scott and Seymour confirmed a conjecture of Gyarfas and proved that if a graph G has no odd holes then chi(G) <=( 2 omega(G)+2). Chudnovsky, Robertson, Seymour and Thomas showed that if G has neither K-4 nor odd holes then chi(G) <= 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 chi(G) <= 4 and chi(G) <= 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 abipartite subgraph. This answers some questions in .
Xu, Baogang; Yu, Gexin; and Zha, Xiaoya, A note on chromatic number and induced odd cycles (2017). ELECTRONIC JOURNAL OF COMBINATORICS, 24(4), 1-8.