On two conjectures of Ho\`ang

TL;DR

This paper disprove Hoàng's conjecture that graphs with independence number at most 3 are perfectly divisible, but confirms that even-hole-free graphs are 3-divisible, advancing understanding of graph partition properties.

Contribution

The paper disproves a conjecture about perfect divisibility in graphs with independence number at most 3 and proves that even-hole-free graphs are 3-divisible.

Findings

Disproved Hoàng's conjecture on perfect divisibility for graphs with independence number ≤ 3.

Confirmed that even-hole-free graphs are 3-divisible.

Established bounds on chromatic number related to divisibility properties.

Abstract

A graph $G$ is said to be perfectly divisible if for every induced subgraph $H$ of $G$ with at least one edge, the vertex set $V(H)$ can be partitioned into two sets $A, B$ such that $H[A]$ is perfect and $\omega(B) < \omega(H)$. It is easy to see that the chromatic number of a perfectly divisible graph is at most $\binom{\omega(G)+1}{2}$. Ho\`ang conjectured that every graph $G$ with $\alpha(G) \le 3$ is perfectly divisible. We disprove this conjecture. In the same vein, a graph $G$ with at least one edge is $k$-divisible if for every induced subgraph $H$ of $G$ with at least one edge, the vertex set $V(H)$ can be partitioned into $k$ sets, none of which contains a largest clique of $H$. It is easy to see that the chromatic number of a $k$-divisible graph is at most $k^{\omega-1}$. Ho\`ang conjectured that every even-hole-free graph is 3-divisible. We confirm this conjecture.