  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.
