# Perfect divisibility and 2-divisibility

## TL;DR

This paper investigates special divisibility properties of graphs, proving that certain classes of graphs without specific induced subgraphs are either 2-divisible or perfectly divisible, revealing structural insights.

## Contribution

It establishes new divisibility properties for $(P_5,C_5)$-free and bull-free graphs, expanding understanding of graph partitioning related to perfect graphs.

## Key findings

- $(P_5,C_5)$-free graphs are 2-divisible
- Bull-free graphs with odd-hole-free or $P_5$-free conditions are perfectly divisible
- Provides structural characterizations related to graph divisibility

## Abstract

A graph $G$ is said to be $2$-divisible if for all (nonempty) induced subgraphs $H$ of $G$, $V(H)$ can be partitioned into two sets $A,B$ such that $\omega(A) < \omega(H)$ and $\omega(B) < \omega(H)$. A graph $G$ is said to be perfectly divisible if for all induced subgraphs $H$ of $G$, $V(H)$ can be partitioned into two sets $A,B$ such that $H[A]$ is perfect and $\omega(B) < \omega(H)$. We prove that if a graph is $(P_5,C_5)$-free, then it is $2$-divisible. We also prove that if a graph is bull-free and either odd-hole-free or $P_5$-free, then it is perfectly divisible.