Sufficient conditions for semicomplete multipartite digraphs to be Hamiltonian

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 (J. Graph Theory 24 (1997) 175-185), proved that every regular semicomplete multipartite digraph is Hamiltonian. With this, Yeo confirmed a conjecture of Zhang (Ann. Discrete Math. 41 (1989) 499-514). In the first part of this paper, a generalization of regularity is considered. We extend Yeo's result to semicomplete multipartite digraphs that satisfy this generalized condition apart from exactly two exceptions. In the second part, we introduce the so-called semi-partition complete 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.