Perfect divisibility and perfect-Pollyanna in bull-free graphs

Abstract

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 $\{P_{17},C_6,C_5,C_4\}$.