On cycles in regular 3-partite tournaments Original Research Article

The vertex set of a digraph D is denoted by V(D)V(D). A c-partite tournament is an orientation of a complete c-partite graph. In 1991, Jian-zhong Wang conjectured that every arc of a regular 3-partite tournament D is contained in directed cycles of all lengths 3,6,9,…,|V(D)|3,6,9,…,|V(D)|. This conjecture is not valid, because for each integer t with 3⩽t⩽|V(D)|3⩽t⩽|V(D)|, there exists an infinite family of regular 3-partite tournaments D such that at least one arc of D is not contained in a directed cycle of length t.