  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 $\{2K_2,
K_5-e\}$-free graphs. Finally, for $p\geq3$, we find polynomial $\chi$-binding
functions for $\{pK_2, H\}$-free graphs where $H\in \{gem, diamond, K_2+P_4,
HVN, K_5-e, butterfly, gem^+, dart, K_1 + C_4, C_4, \overline{P_5}\}$.
