The chromatic number of ($P_5,K_5-e$)-free graphs

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 $\chi (G)\le f(\omega (G))$ 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: $\chi (G)\le \max \{13,\omega (G)+1\}$.