Towards Optimal χ-Binding Functions of (2K1 ∨ K2)-Free Graphs and (P3 ∨ K1)-Free Graphs

A function f : N -> R is called a chi-binding function for a hereditary family G of graphs, if chi(G) <= f (omega(G)) for every G is an element of G where chi (G) and omega (G) denote the chromatic number and clique number respectively. In his influential work, Gyarfas (1987) showed that the family of (2K(1 )boolean OR K-2)-free graphs and the family of (P-3 boolean OR K-1)-free graphs are chi-bounded. Randerath and Schiermeyer (2004) improved the chi-binding functions of both these classes to ((x+1)(2)). In this paper, we further improve the chi-binding function of both these classes to x(2)/2 for x >= 3. Furthermore, we obtain a tight chromatic bound for (P-3 boolean OR K-1)-free graphs with clique number 4.