The χ -Boundedness of P 2 ∪ P 3 -Free Graphs

Abstract

In the early 1980s, Gyárfás introduced the concept of the $\chi$ -bound with $\chi$ -binding functions thereby extending the notion of perfectness. There are a number of challenging conjectures about the $\chi$ -bound. Let $\chi \left(G\right)$ , $\omega \left(G\right)$ , and $\mathrm{\Delta }\left(G\right)$ be the chromatic number, clique number, and maximum degree of a graph $G$ , respectively. In this paper, we prove that if $G$ is a triangle-free and $\left(P_2\cup P_3\right)$ -free graph, then $\chi \left(G\right)\le 3$ unless $G$ is one of eight graphs with $\mathrm{\Delta }\left(G\right)=5$ and $\chi \left(G\right)=4$ , where the eight graphs are extended from the Grötzsch graph as a Mycielskian of a 5-cycle graph. Moreover, we also show that $\chi \left(G\right)\le 3\omega \left(G\right)$ if $G$ is a $\left\{P_2\cup P_3,W_4\right\}$ -free graph.