Leírás
We proved a theorem jointly with Attila Sali almost twenty years ago which stated that every self-complementary graph can be oriented in such a way that making the complementary graph inherit this orientation and considering the union of the so obtained two oriented graphs the tournament we get is transitive. 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 will be mentioned as well. Joint work with Attila Sali and Gábor Tardos.