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arXiv · 2608.13519

Every fork-free graph is perfectly weight divisible

Feng Liu·Shuang Sun·Yan Wang·Qi Wu·Jiasheng Zeng
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Abstract

A graph $G$ is \emph{perfectly weight divisible} if, for every positive integral weight function on $V(G)$ and every induced subgraph $H$ of $G$ with at least one edge, the vertex set $V(H)$ can be partitioned into two sets $A$ and $B$ such that $H[A]$ is perfect and the maximum weight of a clique in $H[B]$ is smaller than the maximum weight of a clique in $H$. Perfect divisibility and its weighted form provide a natural approach to polynomial $\chi$-boundedness. A \emph{fork}, also known as a \emph{chair}, is the graph obtained from a claw by subdividing one of its edges once. In this paper, we prove that every fork-free graph is perfectly weight divisible. As a consequence, we confirm a conjecture of Sivaraman that every fork-free graph is perfectly divisible.

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Feng Liu, Shuang Sun, Yan Wang, Qi Wu, Jiasheng Zeng. 2026-08-13. Every fork-free graph is perfectly weight divisible. https://arxiv.org/abs/2608.13519

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