A graph $G$ is {\em perfectly divisible} if, for each induced subgraph $H$ of $G$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $ω(H[B])<ω(H)$. A {\em bull} is a graph consisting of a triangle with two disjoint pendant edges. Hoàng [Discrete Math. 349 (2026) 114809] proposed four conjectures: 1. $P_5$-free graphs are perfectly divisible; 2. Odd hole-free graphs are perfectly divisible; 3. Even hole-free graphs are perfectly divisible; and 4. $4K_1$-free graphs are perfectly divisible. Karthick et al. [Electron. J. Combin. 29 (2022) P3.19] proposed a conjecture: Fork-free graphs are perfectly divisible. In this paper, we prove that all of five conjectures above hold for bull-free graphs. Our results also generalize some results of Chudnovsky and Sivaraman [J. Graph Theory 90 (2019) 54--60] and Karthick et al. [Electron. J. Combin. 29 (2022) P3.19]. We say that a class ${\cal C}$ is {\em perfect-Pollyanna} if ${\cal C}\cap {\cal G}$ is perfectly divisible for any hereditary class ${\cal G}$ in which each triangle-free graph is 3-colorable. Let $H\in\{\text{house, hammer, diamond}\}$. In this paper, we prove that the class of $(\text{bull}, H)$-free graphs is perfect-Pollyanna. Let ${\cal C}$ be the class of $(\text{bull}, H)$-free graphs. This implies that ${\cal C}\cap {\cal G}$ is perfectly divisible if and only if all of triangle-free graphs in ${\cal G}$ are perfectly divisible. As corollaries, we show that $(\text{bull},{\cal H})$-free graphs are perfectly divisible, where ${\cal H}$ is one of $\{P_{11},C_4\},\{P_{14},C_5,C_4\}$, and $\{P_{17},C_6,C_5,C_4\}$.

翻译：图$G$被称为**完美可划分**的，如果对于$G$的每个诱导子图$H$，$V(H)$可划分为$A$和$B$，使得$H[A]是完美的且ω(H[B])<ω(H)$。**牛图**是由一个三角形附带两个不相交的悬挂边构成的图。Hoàng [Discrete Math. 349 (2026) 114809] 提出了四个猜想：1. 无$P_5$图是完美可划分的；2. 无奇洞图是完美可划分的；3. 无偶洞图是完美可划分的；4. 无$4K_1$图是完美可划分的。Karthick 等人 [Electron. J. Combin. 29 (2022) P3.19] 提出了一个猜想：无叉图是完美可划分的。本文证明，上述五个猜想均对无牛图成立。我们的结果还推广了 Chudnovsky 和 Sivaraman [J. Graph Theory 90 (2019) 54--60] 以及 Karthick 等人 [Electron. J. Combin. 29 (2022) P3.19] 的部分结论。我们称一个类${\cal C}$是**完美乐观的**，如果对于任意使得每个无三角形图均为3-可着色的遗传类${\cal G}$，有${\cal C}\cap {\cal G}$是完美可划分的。设$H\in\{\text{house, hammer, diamond}\}$。本文证明，$(\text{bull}, H)$-无图类是完美乐观的。设${\cal C}$为$(\text{bull}, H)$-无图类，这意味着${\cal C}\cap {\cal G}$是完美可划分的当且仅当${\cal G}$中所有无三角形图都是完美可划分的。作为推论，我们证明$(\text{bull},{\cal H})$-无图是完美可划分的，其中${\cal H}$分别为$\{P_{11},C_4\}$、$\{P_{14},C_5,C_4\}$和$\{P_{17},C_6,C_5,C_4\}$之一。