Perfect divisibility and 2-divisibility

A graph G is said to be 2-divisible if for all (nonempty) induced subgraphs H of G, can be partitioned into two sets such that and . (Here 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, can be partitioned into two sets such that is perfect and . We prove that if a graph is -free, then it is 2-divisible. We also prove that if a graph is bull-free and either odd-hole-free or P₅-free, then it is perfectly divisible.