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Let $G$ and $H$ be two vertex disjoint graphs. The {\em union} $G\cup H$ isthe graph with $V(G\cup H)=V(G)\cup V(H)$ and $E(G\cup H)=E(G)\cup E(H)$. Weuse $P_k$ to denote a {\em path} on $k$ vertices, use {\em house} to denote thecomplement 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$\omega(G)\ge2$.

Optimal chromatic bound for ($P_3\cup P_2$, house)-free graphs