  Let $F_1$ and $F_2$ be two disjoint graphs. The union $F_1\cup F_2$ is a
graph with vertex set $V(F_1)\cup V(F_2)$ and edge set $E(F_1)\cup E(F_2)$, and
the join $F_1+F_2$ is a graph with vertex set $V(F_1)\cup V(F_2)$ and edge set
$E(F_1)\cup E(F_2)\cup \{xy\;|\; x\in V(F_1)\mbox{ and } y\in V(F_2)\}$. In
this paper, we present a characterization to $(P_5, K_1\cup K_3)$-free graphs,
prove that $\chi(G)\le 2\omega(G)-1$ if $G$ is $(P_5, K_1\cup K_3)$-free. Based
on this result, we further prove that $\chi(G)\le $max$\{2\omega(G),15\}$ if
$G$ is a $(P_5,K_1+( K_1\cup K_3))$-free graph, and construct an infinite
family of $(P_5, K_1+( K_1\cup K_3))$-free graphs such that every graph $G$ in
the family satisfies $\chi(G)=2\omega(G)$.
