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

Let $G$ be a graph. We use $\chi(G)$ and $\omega(G)$ 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 $\chi(G)\leq \max\{\min\{16, \omega(G)+3\}, \omega(G)+1\}$. This upper bound is almost sharp.