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

TL;DR

This paper investigates perfect divisions in specific classes of graphs, proving their existence under certain conditions and providing new proofs for related graph classes, advancing understanding of graph partitioning properties.

Contribution

It establishes that ($P_2 rac14 P_4$, bull)-free graphs with no homogeneous set and clique number at least 3 are perfectly divisible, and offers a concise proof for ($P_5$, bull)-free graphs.

Findings

($P_2 rac14 P_4$, bull)-free graphs with  no homogeneous set are perfectly divisible

Counterexample exists for  

Short proof provided for ($P_5$, bull)-free graphs' perfect divisibility

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 $\omega(G) \geq 3$ 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.].