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

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 $K4$ . Let $H$ be the complement of a cycle on 7 vertices. Chudnovsky et al [6] proved that every (odd hole, $K4$)-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 $G$, $\chi(G)\le \omega(G)+1$, and the equality holds if and only if $\omega(G)=3$ and $G$ has $H$ as an induced subgraph.