Chromatic bounds for the subclasses of $p{K}_2$-free graphs

The concept of $\chi$-binding functions for classes of free graphs has been extensively studied in the past. In this paper, we improve the existing $\chi$-binding function for $\{2K_2,K_1+C_4\}$-free graphs. Also, we find a linear $\chi$-binding function for $\{2K_2,K_2+P_4\}$-free graphs. In addition, we give alternative proofs for the $\chi$-binding function of $\{2K_2,gem\}$-free graphs, $\{2K_2,HVN\}$-free graphs and ${2K_2, K_5-e}$-freegraphs\mathrm{.}Finally,for$p\geq3$,wefindpolynomial$\chi$-binding functions for $\{p{K}_2,H\}$-free graphs where $H\in {gem, diamond, K_2+P_4, HVN, K_5-e, butterfly, gem^+, dart, K_1 + C_4, C_4, \overline{P_5}}$.