On Graphs With No Induced P 5 or K 5 − e

Abstract

In this paper, we are interested in some problems related to chromatic number and clique number for the class of $\left(P_5,K_5-e\right)$-free graphs and prove the following the results: (a) If $G$ is a connected ($P_5,K_5-e$)-free graph with $\omega \left(G\right)\ge 7$, 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 $\omega \left(H\right)=6$ and has no clique cut-set. This strengthens a result of Malyshev and Lobanova (Discrete Applied Mathematics 219 [2017] 158–166). (b) If $G$ is a ($P_5,K_5-e$)-free graph with $\omega \left(G\right)\ge 4$, then $\chi \left(G\right)\le \max \left\{7,\omega \left(G\right)\right\}$. Moreover, the bound is tight when $\omega \left(G\right)\notin \left\{4,5,6\right\}$. This result, together with known results, partially answers a question of Ju and Huang (Theoretical Computer Science 993 [2024] Article No.: 114465) and also improves a result of Xu [Manuscript 2022]. While Chromatic Number is known to be $NP$-hard for the class of $P_5$-free graphs, our results, together with some known results, imply that Chromatic Number can be solved in polynomial time for the class of ($P_5,K_5-e$)-free graphs, which may be of independent interest.