On near optimal colorable graphs

A class of graphs $\mathcal{G}$ is said to be \emph{near optimal colorable} if there exists a constant $c\in \mathbb{N}$ such that every graph $G\in \mathcal{G}$ satisfies $\chi (G)\le \max \{c,\omega (G)\}$, where $\chi (G)$ and $\omega (G)$ respectively denote the chromatic number and clique number of $G$. The class of near optimal colorable graphs is an important subclass of the class of $\chi$-bounded graphs which is well-studied in the literature. In this paper, we show that the class of ($F,K_4-e$)-free graphs is near optimal colorable, where $F\in \{P_1+2P_2,2P_1+P_3,3P_1+P_2\}$ and the graph $K_4-e$ is commonly referred as the {\em diamond}. This partially answers a question of Ju and Huang [Theoretical Computer Science 993 (2024) Article No.: 114465] and is related to a question of Schiermeyer (unpublished). Furthermore, using these results with some earlier known results, we also provide an alternate proof to the fact that the $\text{textsc}ChromaticNumber$ problem for the class of ($F,K_4-e$)-free graphs is solvable in polynomial time, where $F\in \{P_1+2P_2,2P_1+P_3,3P_1+P_2\}$.