Optimal chromatic bound for ($P_2\cup P_4$, HVN)-free graphs

The HVN is a graph formed by removing two edges incident to the same vertex from the complete graph $K_5$. In this paper, we prove that every ($P_2\cup P_4$, HVN)-free graph $G$ satisfies $\chi (G)\le \lceil \frac{4}{3}\omega (G)\rceil$ when $\omega (G)\ge 4$, where $\chi (G)$ and $\omega (G)$ denote the chromatic number and clique number of $G$, respectively. Furthermore, this bound is optimal for every $\omega (G)\ge 4$. Constructions demonstrating the optimality of the bound are provided. Our work unifies several previously known results on $\chi$-binding functions for several graph classes.