On the chromatic number of some ($P_3\cup P_2$)-free graphs

TL;DR

This paper investigates bounds on the chromatic number for certain ($P_3rac12$)free graphs, establishing new upper bounds for classes defined by additional forbidden subgraphs, and discusses $$-binding functions for specific graph classes.

Contribution

It provides new upper bounds on the chromatic number for ($P_3rac12$)-free graphs with additional forbidden subgraphs, advancing understanding of $$-boundedness in these classes.

Findings

$ heta(G)$ bound for ($P_3rac12$, kite)-free graphs

$ heta^2(G)$ bound for ($P_3rac12$, hammer)-free graphs

$(3 heta^2(G)+ heta(G))/2$ bound for ($P_3rac12$, $C_5$)-free graphs

Abstract

A hereditary class $\cal G$ of graphs is {\em $\chi$-bounded} if there is a {\em $\chi$-binding function}, say $f$, such that $\chi(G)\le f(\omega(G))$ for every $G\in\cal G$, where $\chi(G)(\omega(G))$ denotes the chromatic (clique) number of $G$. It is known that for every $(P_3\cup P_2)$-free graph $G$, $\chi(G)\le \frac{1}{6}\omega(G)(\omega(G)+1)(\omega(G)+2)$ \cite{BA18}, and the class of $(2K_2, 3K_1)$-free graphs does not admit a linear $\chi$-binding function\cite{BBS19}. In this paper, we prove that (\romannumeral 1) $\chi(G)\le2\omega(G)$ if $G$ is ($P_3\cup P_2$, kite)-free, (\romannumeral 2) $\chi(G)\le\omega^2(G)$ if $G$ is ($P_3\cup P_2$, hammer)-free, (\romannumeral 3) $\chi(G)\le\frac{3\omega^2(G)+\omega(G)}{2}$ if $G$ is ($P_3\cup P_2, C_5$)-free. Furthermore, we also discuss $\chi$-binding functions for $(P_3\cup P_2, K_4)$-free graphs.