On cycles in regular 3-partite tournaments

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.In this paper, we prove that every arc of a regular 3-partite tournament with at least nine vertices is contained in a directed cycle of length m  , m+1m+1, or m+2m+2 for 3⩽m⩽53⩽m⩽5, and we conjecture that every arc of a regular 3-partite tournament is contained in a directed cycle of length m  , (m+1)(m+1), or (m+2)(m+2) for each m∈{3,4,…,|V(D)|-2}m∈{3,4,…,|V(D)|-2}.It is known that every regular 3-partite tournament D   with at least six vertices contains directed cycles of lengths 3, |V(D)|-3|V(D)|-3, and |V(D)||V(D)|. We show that every regular 3-partite tournament D   with at least six vertices also has a directed cycle of length 6, and we conjecture that each such 3-partite tournament contains cycles of all lengths 3,6,9,…,|V(D)|3,6,9,…,|V(D)|.