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A note on chromatic number and induced odd cycles
Xu, Baogang Yu, Gexin Zha, Xiaoya
Xu, Baogang
Yu, Gexin
Zha, Xiaoya
Abstract
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 [17].
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2017-11-03
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Mathematics