On graphs with no induced P_5 or K_5-e
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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 ω(G)≥ 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 ω(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 ω(G)≥ 4, then χ(G)≤max{7, ω(G)}. Moreover, the bound is tight when ω(G)∉{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.
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