Perfect divisibility and 2-divisibility

A graph G is said to be 2-divisible if for all (nonempty) induced subgraphs H of G, V(H) can be partitioned into two sets A,B such that omega(A)<omega(H) and omega(B)<omega(H). (Here omega(G) denotes the clique number of G, the number of vertices in a largest clique of G). A graph G is said to be perfectly divisible if for all induced subgraphs H of G, V(H) can be partitioned into two sets A,B such that H[A] is perfect and omega(B)<omega(H). We prove that if a graph is (P5,C5)-free, then it is 2-divisible. We also prove that if a graph is bull-free and either odd-hole-free or P-5-free, then it is perfectly divisible.