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

TL;DR

This paper proves a tight linear upper bound on the chromatic number for ($P_3 P_2, W_4$)-free graphs, improving previous results and extending known bounds.

Contribution

It establishes a tight bound of (G) 2 (G) for a new class of graphs, extending prior work and generalizing existing results.

Findings

(G) (G) for ($P_3 P_2, W_4$)-free graphs

Bound is tight for =2,3

Improves previous bounds by Wang and Zhang

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.}.