Finding a longest path in a complete multipartite digraph

A digraph obtained by replacing each edge of a complete $m$-partite graph with an arc or a pair of mutually opposite arcs with the same end vertices is called a complete $m$-partite digraph. An $O ( n^3 )$ algorithm for finding a longest path in a complete $m$-partite $( m \geq 2 )$ digraph with $n$ vertices is described in this paper. The algorithm requires time $O( n^{2.5} )$ in case of testing only the existence of a Hamiltonian path and finding it if one exists. It is simpler than the algorithm of Manoussakis and Tuza [SIAM J. Discrete Math., 3 (1990), pp. 537–543], which works only for $m = 2$. The algorithm implies a simple characterization of complete $m$-partite digraphs having Hamiltonian paths that was obtained for the first time in Gutin [Kibernetica (Kiev), 4 (1985), pp. 124–125] for $m = 2$ and in Gutin [Kibernetica (Kiev), 1(1988), pp. 107–108] for $ m \geq 2 $.