$\chi$-binding functions for some classes of $(P_3\cup P_2)$-free graphs

The class of $2K_2$-free graphs have been well studied in various contexts in the past. It is known that the class of $\{2K_2,2K_1+K_p\}$-free graphs and $\{2K_2,(K_1\cup K_2)+K_p\}$-free graphs admits a linear $\chi$-binding function. In this paper, we study the classes of $(P_3\cup P_2)$-free graphs which is a superclass of $2K_2$-free graphs. We show that $\{P_3\cup P_2,2K_1+K_p\}$-free graphs and $\{P_3\cup P_2,(K_1\cup K_2)+K_p\}$-free graphs also admits linear $\chi$-binding functions. In addition, we give tight chromatic bounds for $\{P_3\cup P_2,HVN\}$-free graphs and $\{P_3\cup P_2,diamond\}$-free graphs and it can be seen that the latter is an improvement of the existing bound given by A. P. Bharathi and S. A. Choudum [Colouring of $(P_3\cup P_2)$-free graphs, Graphs and Combinatorics 34 (2018), 97-107].