  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.
