  Let $G$ be a graph. We use $\chi(G)$ and $\omega(G)$ to denote the chromatic
number and clique number of $G$ respectively. A $P_5$ is a path on 5 vertices.
A family of graphs $\mathcal{G}$ is said to be {\it$\chi$-bounded} if there
exists some function $f$ such that $\chi(G)\leq f(\omega(G))$ for every
$G\in\mathcal{G}$. In this paper, we show that the family of $(P_5,
K_5-e)$-free graphs is $\chi$-bounded by a linear function: $\chi(G)\leq
\max\{13,\omega(G)+1\}$.
