Finding a Longest Path in a Complete Multipartite Digraph

A digraph obtained by replacing each edge of a complete mpartite graph with an arc or a pair of mutually opposite arcs with the same end vertices is called a complete m-partite digraph. We describe an O(n) algorithm for finding a longest path in a complete m-partite (m ≥ 2) digraph with n vertices. The algorithm requires time O(n) 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 [4], which works only for m = 2. Our algorithm implies a simple characterization of complete m-partite digraphs having Hamiltonian paths which was obtained for the first time in [1] (for m = 2) and in [2] (for m ≥ 2).