Counterexamples to a Conjecture of Wang Concerning Regular 3-partite Tournaments

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 \( 3, 6, 9, \ldots, |V(D)| \).

In this paper, we show that this conjecture is completely false. Namely, for each integer \( t \) with \( 3 \leq t \leq |V(D)| \), we present an infinite family of regular 3-partite tournaments \( D \) such that there exists an arc in \( D \) which is not contained in a directed cycle of length \( t \).