Suucient Conditions for Semicomplete Multipartite Digraphs to Be Hamiltonian Dedicated to Professor Dr. Horst Sachs on His 70th Birthday

A semicomplete multipartite digraph is obtained by replacing each edge of a complete multipartite graph by an arc or by a pair of two mutually opposite arcs. Very recently, Yeo 7] proved that every regular semicomplete multipartite digraph is Hamiltonian. With this, Yeo connrmed a conjecture of C.-Q. Zhang 8]. In the rst part of this paper, a generalization of regularity is considered. We extend Yeo's result to semicomplete multipartite digraphs that satisfy this condition apart from exactly two exceptions. In the second part, we introduce so called semi-partitioncomplete digraphs and show that this family is Hamiltonian or cycle complementary, when, clearly, the cardinality of each partite set is less than or equal to half the order.