  For a graph $G$, $\chi(G)$ $(\omega(G))$ denote its chromatic (clique)
number. A $P_5$ is the chordless path on five vertices, and a $4$-$wheel$ is
the graph consisting of a chordless cycle on four vertices $C_4$ plus an
additional vertex adjacent to all the vertices of the $C_4$. In this paper, we
show that every ($P_5$, $4$-wheel)-free graph $G$ satisfies $\chi(G)\leq
\frac{3}{2}\omega(G)$. Moreover, this bound is almost tight. That is, there is
a class of ($P_5$, $4$-wheel)-free graphs $\cal L$ such that every graph $H\in
\cal L$ satisfies $\chi(H)\geq\frac{10}{7}\omega(H)$. This generalizes/improves
several previously known results in the literature.
