每个无fork图都是完美权可分的
Every fork-free graph is perfectly weight divisible
AI总结：
本文证明了无fork图均满足完美权可分性，进而证实了Sivaraman关于无fork图完美可分性的猜想，为多项式χ-有界性研究提供了新结论。
AI中文摘要：
若图$G$满足：对$V(G)$上的每个正整数权函数，以及$G$的每个至少含一条边的诱导子图$H$，顶点集$V(H)$均可划分为两个集合$A$和$B$，使得$H[A]$是完美图，且$H[B]$中团的最大权小于$H$中团的最大权，则称$G$是**完美权可分的**。完美可分性及其加权形式为多项式$χ$-有界性提供了一种自然途径。**fork图**（也称为chair图）是将爪图的一条边细分一次得到的图。本文证明了每个无fork图都是完美权可分的。作为推论，我们证实了Sivaraman提出的每个无fork图都是完美可分的猜想。
英文摘要：
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 $χ$-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.
