Spanning Multipartite Tournaments of Semicomplete Multipartite Digraphs

A digraph obtained by replacing each edge of a complete \(n\)-partite graph by an arc or a pair of mutually opposite arcs is called a semi-complete \(n\)-partite digraph. An \(n\)-partite tournament is an orientation of a complete \(n\)-partite graph. In this paper we shall prove that a strongly connected semicomplete \(n\)-partite digraph with a longest directed cycle \(C\), contains a spanning strongly connected \(n\)-partite tournament which also has the longest directed cycle \(C\) with exception of a well determined family of semicomplete bipartite digraphs. This theorem shows that many well-known results on strongly connected \(n\)-partite tournaments are also valid for strongly connected semicomplete \(n\)-partite digraphs.