Perfect divisions in ($P_2\cup P_4$, bull)-free graphs

Abstract

A graph $G$ has a perfect division if its vertex set can be partitioned into two sets $A$, $B$ such that $G[A]$ is perfect and $\omega (G[B])<\omega (G)$. We call $G$ perfectly divisible if every induced subgraph of $G$ admits a perfect division. We prove that every ($P_2\cup P_4$, bull)-free graph $G$ with $\omega (G)\ge 3$ has a perfect division if $G$ contains no homogeneous set. The clique-number condition is tight: a counterexample exists for $\omega (G)=2$. Additionally, we present a short proof of the perfect divisibility of ($P_5$, bull)-free graphs, originally established by Chudnovsky and Sivaraman [J. Graph Theory 90 (2019), 54-60.].