Chromatic bounds for some classes of $2K_2$-free graphs

A hereditary class $\mathcal{G}$ of graphs is $\chi$-bounded if there is a $\chi$-binding function, say $f$ such that $\chi(G) \leq f(\omega(G))$, for every $G \in \cal{G}$, where $\chi(G)$ ($\omega(G)$) denote the chromatic (clique) number of $G$. It is known that for every $2K2$-free graph $G$, $\chi(G) \leq \binom{\omega(G)+1}{2}$, and the class of ($2K2, 3K1$)-free graphs does not admit a linear $\chi$-binding function. In this paper, we are interested in classes of $2K2$-free graphs that admit a linear $\chi$-binding function. We show that the class of ($2K2, H$)-free graphs, where $H\in {K1+P4, K1+C4, \overline{P2\cup P3}, HVN, K5-e, K5}$ admits a linear $\chi$-binding function. Also, we show that some superclasses of $2K2$-free graphs are $\chi$-bounded.