  Let $G$ and $H$ be two vertex disjoint graphs. The {\em union} $G\cup H$ is
the graph with $V(G\cup H)=V(G)\cup V(H)$ and $E(G\cup H)=E(G)\cup E(H)$. We
use $P_k$ to denote a {\em path} on $k$ vertices, use {\em house} to denote the
complement of $P_5$. In this paper, we show that $\chi(G)\le2\omega(G)$ if $G$
is ($P_3\cup P_2$, house)-free. Moreover, this bound is optimal when
$\omega(G)\ge2$.
