Transitive orientations in bull-reducible Berge graphs

A bull is a graph with five vertices r,y,x,z,sr,y,x,z,s and five edges ryry, yxyx, yzyz, xzxz, zszs. A graph GG is bull-reducible if every vertex of GG lies in at most one bull of GG. We prove that every bull-reducible Berge graph GG that contains no antihole is weakly chordal, or has a homogeneous set, or is transitively orientable. This yields a fast polynomial time algorithm to color the vertices of such a graph exactly.