A sufficient condition for a semicomplete multipartite digraph to be Hamiltonian

A digraph obtained by replacing each edge of a complete n-partite (n 2:: 2) graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicomplete n-partite digraph or semicomplete multipartite digraph (abbreviated to SMD). In this paper we show the following result for a semicomplete multipartite digraph of order p with the partite sets VI, 112, ... , Vn. Let r = minl<i<n{l"vil}. Hfor each pair of dominated nonadjacent vertices {x, y}, d(x)+d(y) 2:: min{2(p-r)+3, 2p-1}, then T is Hamiltonian. This result is best possible in a sense.