The chromatic number of (P_5, HVN )-free graphs

Let $G$ be a graph. We use $χ(G)$ and $ω(G)$ to denote the chromatic number and clique number of $G$ respectively. A $P_5$ is a path on 5 vertices, and an $HVN$ is a $K_4$ together with one more vertex which is adjacent to exactly two vertices of $K_4$. Combining with some known result, in this paper we show that if $G$ is $(P_5, \textit{HVN})$-free, then $χ(G)\leq \max\{\min\{16, ω(G)+3\}, ω(G)+1\}$. This upper bound is almost sharp.

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