An optimal chromatic bound for the class of {P3∪2K1,P3∪2K1¯}-free graphs

Abstract

In 1987, A. Gyárfás in his paper “Problems from the world surrounding perfect graphs” posed the problem of determining the smallest $\chi$-binding function for $G(F,\overline{F})$, when $G\left(F\right)$ is $\chi$-bounded. So far the problem has only been attempted for some forest $F$ with four or five vertices. In this paper, we address the problem when $F=P_3\cup 2K_1$ and show that if $G$ is a $\{P_3\cup 2K_1,\overline{P_3\cup 2K_1}\}$-free graph with $\omega \left(G\right)\ne 3$, then it admits $\omega \left(G\right)+1$ as a $\chi$-binding function. Moreover, we also construct examples to show that this bound is tight for all values of $\omega \ne 3$.