On cycles in regular 3-partite tournaments

Abstract

The vertex set of a digraph D is denoted by 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 ) | . This conjecture is not valid, because for each integer t with 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 + 1 , or m + 2 for 3 ⩽ 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 ) , or ( m + 2 ) for each 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 , and | 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 ) | .