Linear χ-binding functions for some classes of (P3∨P2)-free graphs

The class of 2K(2)-free graphs has been well studied in the past. In the paper "On the chromatic number of 2K(2)-free graphs, Discrete Applied Mathematics, 253 (2019), 14-24", it was shown that the class of {2K(2),2K(1)+Kp}-free graphs and {2K(2),(K-1 boolean OR K-2)+Kp}-free graphs admit a linear chi-binding function. In this paper, we study some subclasses of (P-3 boolean OR P-2)-free graphs which is a superclass of 2K(2)-free graphs. We show that {P-3 boolean OR P(2,)2K(1)+K-p}-free graphs and {P3 boolean OR P2,(K-1 boolean OR K-2)+K-p}-free graphs also admit linear chi-binding functions. In addition, we give a tight chi-binding function for {P-3 boolean OR P-2,HVN}-free graphs and improve the chi-bound for {P-3 boolean OR P-2,diamond}-free graphs with omega = 4.