Longest cycles in almost regular 3-partite tournaments

Abstract

If D is a digraph, then we denote by V ( D ) its vertex set. A multipartite or c -partite tournament is an orientation of a complete c -partite graph. The global irregularity of a digraph D is defined by i g ( D ) = max { max ( d + ( x ) , d - ( x ) ) - min ( d + ( y ) , d - ( y ) ) | x , y ∈ V ( D ) } . If i g ( D ) = 0 , then D is regular, and if i g ( D ) ⩽ 1 , then D is called almost regular. In 1997, Yeo has shown that each regular multipartite tournament is Hamiltonian. This remains valid for almost all almost regular c -partite tournaments with c ≥ 4 . However, there exist infinite families of almost regular 3-partite tournaments without any Hamiltonian cycle. In this paper we will prove that every vertex of an almost regular 3-partite tournament D is contained in a directed cycle of length at least | V ( D ) | - 2 . Examples will show that this result is best possible.