  For two vertex disjoint graphs $H$ and $F$, we use $H\cup F$ to denote the
graph with vertex set $V(H)\cup V(F)$ and edge set $E(H)\cup E(F)$, and use
$H+F$ to denote the graph with vertex set $V(H)\cup V(F)$ and edge set
$E(H)\cup E(F)\cup\{xy\;|\; x\in V(H), y\in V(F)$$\}$. A $W_4$ is the graph
$K_1+C_4$. In this paper, we prove that $\chi(G)\le 2\omega(G)$ if $G$ is a
($P_3\cup P_2, W_4$)-free graph. This bound is tight when $\omega =2$ and $3$,
and improves the main result of Wang and Zhang. Also, this bound partially
generalizes some results of Prashant {\em et al.}.
