Perfect divisibility and coloring of some fork-free graphs

A hole is an induced cycle of length at least four, and an odd hole is a hole of odd length. A fork is a graph obtained from K1,3 by subdividing an edge once. An odd balloon is a graph obtained from an odd hole by identifying respectively two consecutive vertices with two leaves of K1,3. A gem is a graph that consists of a P4 plus a vertex adjacent to all vertices of the P4. A butterfly is a graph obtained from two triangles 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 ω(H[B])