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.

翻译：本文研究$(P_5,K_5-e)$-自由图类中与色数和团数相关的一些问题，并证明以下结论：$(a)$ 若$G$是连通的$(P_5,K_5-e)$-自由图且$\omega(G)\geq 7$，则要么$G$是二部图的补图，要么$G$具有团割集。此外，存在一个连通的$(P_5,K_5-e)$-自由非完美图$H$满足$\omega(H)=6$且无团割集。这一结果加强了Malyshev和Lobanova [Disc. Appl. Math. 219 (2017) 158--166] 的结论。$(b)$ 若$G$是$(P_5,K_5-e)$-自由图且$\omega(G)\geq 4$，则$\chi(G)\leq \max\{7, \omega(G)\}$。此外，当$\omega(G)\notin \{4,5,6\}$时该界是紧的。该结果结合已有结论部分回答了Ju和Huang [arXiv:2303.18003 [math.CO] 2023] 的问题，同时改进了Xu [手稿 2022] 的结果。尽管已知"色数问题"对$P_5$-自由图类是NP-难的，我们的结果结合一些已知结论表明，"色数问题"可以在多项式时间内求解$(P_5,K_5-e)$-自由图类，这一结论可能具有独立的研究价值。