  In an earlier paper the first two authors have shown that self-complementary
graphs can always be oriented in such a way that the union of the oriented
version and its isomorphically oriented complement gives a transitive
tournament. We investigate the possibilities of generalizing this theorem to
decompositions of the complete graph into three or more isomorphic graphs. We
find that a complete characterization of when an orientation with similar
properties is possible seems elusive. Nevertheless, we give sufficient
conditions that generalize the earlier theorem and also imply that
decompositions of odd vertex complete graphs to Hamiltonian cycles admit such
an orientation. These conditions are further generalized and some necessary
conditions are given as well.
