  In this paper, we are interested in some problems related to chromatic number
and clique number for the class of $(P_5,K_5-e)$-free graphs, and prove the
following. $(a)$ If $G$ is a connected ($P_5,K_5-e$)-free graph with
$\omega(G)\geq 7$, then either $G$ is the complement of a bipartite graph or
$G$ has a clique cut-set. Moreover, there is a connected ($P_5,K_5-e$)-free
imperfect graph $H$ with $\omega(H)=6$ and has no clique cut-set. This
strengthens a result of Malyshev and Lobanova [Disc. Appl. Math. 219 (2017)
158--166]. $(b)$ If $G$ is a ($P_5,K_5-e$)-free graph with $\omega(G)\geq 4$,
then $\chi(G)\leq \max\{7, \omega(G)\}$. Moreover, the bound is tight when
$\omega(G)\notin \{4,5,6\}$. This result together with known results partially
answers a question of Ju and Huang [arXiv:2303.18003 [math.CO] 2023], and also
improves a result of Xu [Manuscript 2022].
  While the "Chromatic Number Problem" is known to be $NP$-hard for the class
of $P_5$-free graphs, our results together with some known results imply that
the "Chromatic Number Problem" can be solved in polynomial time for the class
of ($P_5,K_5-e$)-free graphs which may be independent interest.
