A tight linear chromatic bound for ($P_3\cup P_2,W_4$)-free graphs

Abstract

For two vertex disjoint graphs $H$ and $F$, we use $H\cup F$ to denote the graph with vertex set $V(H)\cup V(F)$ and edge set $E(H)\cup E(F)$, and use $H+F$ to denote the graph with vertex set $V(H)\cup V(F)$ and edge set $E(H)\cup E(F)\cup \{xy|x\in V(H),y\in V(F)$$\}$. A $W_4$ is the graph $K_1+C_4$. In this paper, we prove that $\chi (G)\le 2\omega (G)$ if $G$ is a ($P_3\cup P_2,W_4$)-free graph. This bound is tight when $\omega =2$ and $3$, and improves the main result of Wang and Zhang. Also, this bound partially generalizes some results of Prashant {\em et al.}.

arXiv admin note: text overlap with arXiv:2308.05442, arXiv:2307.11946