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 3) algorithm for finding a longest path in a complete m-partite (m ≥ 2) digraph with n vertices. 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 [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)

On a cyclic connectivity property of directed graphs

AbstractLet us call a digraph D cycle-connected if for every pair of vertices u,v∈V(D) there exists a cycle containing both u and v. In this paper we study the following open problem introduced by Ádám. Let D be a cycle-connected digraph. Does there exist a universal edge in D, i.e., an edge e∈E(D) such that for every w∈V(D) there exists a cycle C such that w∈V(C) and e∈E(C)?In his 2001 paper Hetyei conjectured that cycle-connectivity always implies the existence of a universal edge. In the present paper we prove the conjecture of Hetyei for bitournaments