  For a graph $G$, let $\chi(G)$ ($\omega(G)$) denote its chromatic (clique)
number. A $P_2+P_3$ is the graph obtained by taking the disjoint union of a
two-vertex path $P_2$ and a three-vertex path $P_3$. A $\bar{P_2+P_3}$ is the
complement graph of a $P_2+P_3$. In this paper, we study the class of
($P_2+P_3$, $\bar{P_2+P_3}$)-free graphs and show that every such graph $G$
with $\omega(G)\geq 3$ satisfies $\chi(G)\leq \max \{\omega(G)+3,
\lfloor\frac{3}{2} \omega(G) \rfloor-1 \}$. Moreover, the bound is tight.
Indeed, for any $k\in {\mathbb N}$ and $k\geq 3$, there is a ($P_2+P_3$,
$\bar{P_2+P_3}$)-free graph $G$ such that $\omega(G)=k$ and $\chi(G)=\max\{k+3,
\lfloor\frac{3}{2} k \rfloor-1 \}$.
