Notes on cycles through a vertex or an arc in regular 3-partite tournaments

We shall assume that the reader is familiar with standard terminology on directed graphs (see, e.g., Bang-Jensen and Gutin [1]). In this note, if we speak of a cycle, then we mean a directed cycle. If xy is an arc of a digraph D, then we write x → y and say x dominates y. If X and Y are two disjoint vertex sets of a digraph D such that every vertex of X dominates every vertex of Y , then we say that X dominates Y , denoted by X → Y ; otherwise denoted by X 9 Y . If D is a vertex set or a subdigraph of a digraph D, then we define N D′(x) as the set of vertices of D ′ which are dominated by x and N D′(x) as the set of vertices of D which dominate x. The numbers d+D′(x) = |N + D′(x)| and d − D′(x) = |N − D′(x)| are called the out-degree and in-degree of x in D, respectively. When D = D, N D (x),N − D (x), d + D (x) and d − D (x) are also denoted by N (x),N(x), d(x) and d(x), respectively. For two vertex sets X, Y of a digraph D, we define X − Y = {x|x ∈ X, x ∉ Y }. A c-partite tournament is an orientation of a complete c-partite graph. A digraph D is regular, if d(x) = d(x) = d(y) = d(y) for all x, y ∈ V (D). For cycles in regular 3-partite tournaments, Volkmann [2] obtained the following four results.