($P_2+P_4$, $K_4-e$)-free graphs are nearly $ω$-colorable

Abstract: For a graph $G$, $\chi(G)$ and $\omega(G)$ respectively denote the chromatic number and clique number of $G$. In this paper, we show that if $G$ is a ($P_2+P_4$, $K_4-e$)-free graph with $\omega(G)\geq 3$, then $\chi(G)\leq \max{6, \omega(G)}$, and that the bound is tight for each $\omega(G)\notin {4,5}$. This extends the results known for the class of ($P_2+P_3$, $K_4-e$)-free graphs, improves the bound of Chen and Zhang [arXiv:2412.14524[math.CO], 2024] given for the class of ($P_2+P_4$, $K_4-e$)-free graphs, partially answers a question of Ju and the third author [Theor. Comp. Sci. 993 (2024) Article No.: 114465] on `near optimal colorable graphs', and partially answers a question of Schiermeyer (unpublished) on the chromatic bound for ($P_7$, $K_4-e$)-free graphs.