A $hole$ is an induced cycle of length at least four, and an odd hole is a hole of odd length. A {\em fork} is a graph obtained from $K_{1,3}$ by subdividing an edge once. An {\em odd balloon} is a graph obtained from an odd hole by identifying respectively two consecutive vertices with two leaves of $K_{1, 3}$. A {\em gem} is a graph that consists of a $P_4$ plus a vertex adjacent to all vertices of the $P_4$. A {\em butterfly} is a graph obtained from two traingles by sharing exactly one vertex. A graph $G$ is 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 $\omega(H[B])<\omega(H)$. In this paper, we show that (odd balloon, fork)-free graphs are perfectly divisible (this generalizes some results of Karthick {\em et al}). As an application, we show that $\chi(G)\le\binom{\omega(G)+1}{2}$ if $G$ is (fork, gem)-free or (fork, butterfly)-free.

翻译：一个$hole$是指长度至少为4的诱导圈，而奇hole是长度为奇数的hole。一个{\em fork}是通过将$K_{1,3}$的一条边进行一次细分得到的图。一个{\em odd balloon}是通过将一个奇hole中两个连续顶点分别与$K_{1,3}$的两个叶子顶点等同而得到的图。一个{\em gem}是由一个$P_4$加上一个与该$P_4$所有顶点相邻的顶点构成的图。一个{\em butterfly}是由两个三角形共享恰好一个顶点而得到的图。图$G$被称为完美可除的，如果对于$G$的每个诱导子图$H$，$V(H)$可以划分为$A$和$B$两部分，使得$H[A]$是完美的且$\omega(H[B])<\omega(H)$。在本文中，我们证明了(odd balloon, fork)-free图是完美可除的（这推广了Karthick等人的一些结果）。作为一个应用，我们证明如果$G$是(fork, gem)-free图或(fork, butterfly)-free图，则$\chi(G)\le\binom{\omega(G)+1}{2}$。