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

Abstract

A hereditary class $\mathcal{G}$ 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 $G\in \mathcal{G}$, 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) $\chi (G)\le 2\omega (G)$ 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.

arXiv admin note: substantial text overlap with arXiv:2308.05442, arXiv:2308.08768