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：孔（hole）是指长度至少为4的导出环，奇孔是长度为奇数的孔。叉图（fork）是通过将 $K_{1,3}$ 的一条边进行一次细分得到的图。奇气球图（odd balloon）是由一个奇孔通过将其两个连续顶点分别与 $K_{1,3}$ 的两个叶子顶点等同得到。宝石图（gem）由一个 $P_4$ 加上一个与该 $P_4$ 所有顶点相邻的顶点构成。蝴蝶图（butterfly）由两个三角形共享恰好一个顶点得到。图 $G$ 是完美可分割的，如果对于 $G$ 的每个导出子图 $H$，$V(H)$ 可以划分为 $A$ 和 $B$，使得 $H[A]$ 是完美的且 $\omega(H[B])<\omega(H)$。本文证明，不含奇气球图和叉图的图是完美可分割的（这推广了Karthick等人的一些结果）。作为应用，我们还证明：若 $G$ 不含（叉图, 宝石图）或（叉图, 蝴蝶图），则 $\chi(G)\le\binom{\omega(G)+1}{2}$。