On the chromatic number of a family of odd hole free graphs

10/25/2021

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by   Jialei Song, et al.

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A hole is an induced cycle of length at least 4, and an odd hole is a hole of odd length. A full house is a graph composed by a vertex adjacent to both ends of an edge in K_4 . Let H be the complement of a cycle on 7 vertices. Chudnovsky et al [6] proved that every (odd hole, K_4)-free graph is 4-colorable and is 3-colorable if it does not has H as an induced subgraph. In this paper, we use the proving technique of Chudnovsky et al to generalize this conclusion to (odd hole, full house)-free graphs, and prove that for (odd hole, full house)-free graph G, χ(G)≤ω(G)+1, and the equality holds if and only if ω(G)=3 and G has H as an induced subgraph.