Cycles through a given arc in almost regular multipartite tournaments

If x is a vertex of a digraph D, then we denote by d(x) and d−(x) the outdegree and the indegree of x, respectively. The global irregularity of a digraph D is defined by ig(D) = max{d+(x), d−(x)}−min{d+(y), d−(y)} over all vertices x and y of D (including x = y). If ig(D) = 0, then D is regular and if ig(D) ≤ 1, then D is almost regular. A c-partite tournament is an orientation of a complete c-partite graph. In 1998, Y. Guo showed, if every arc of a regular c-partite tournament is contained in a directed cycle of length 3, then every arc belongs to a directed cycle of length n for each n ∈ {4, 5, . . . , c}. Recently, L. Volkmann generalized this result for c ≥ 6. He showed, if V1, V2, . . . , Vc are the partite sets of an almost regular c-partite tournament with c ≥ 6 and |V1| = |V2| = . . . = |Vc| ≥ 2, then every arc of D is contained in a directed cycle of length n for each n ∈ {4, 5, . . . , c}. In this paper we shall extend this theorem to all almost regular c-partite tournaments with c ≥ 7 such that there are at least two vertices in each partite set. Examples will show that this result is not valid for the case that c = 6 or that c = 7 and there is only one vertex in at least one partite set. 1 Terminology and introduction In this paper all digraphs are finite without loops and multiple arcs. The vertex set and arc set of a digraph D are denoted by V (D) and E(D), respectively. If xy is an arc of a digraph D, then we write x → y and say that x dominates y, and if X and Y are two disjoint vertex sets or subdigraphs of D such that every vertex of X dominates every vertex of Y , then we say that X dominates Y , denoted by X → Y . Furthermore, X Y denotes the fact that there is no arc leading from Y to X . For the number of arcs from X to Y we write d(X, Y ). If D is a digraph, then the outneighborhood N D (x) = N (x) of a vertex x is the set of vertices dominated by x and Australasian Journal of Combinatorics 27(2003), pp.223–245 the in-neighborhood N− D (x) = N −(x) is the set of vertices dominating x. Therefore, if there is an arc xy ∈ E(D), then y is an outer neighbor of x and x is an inner neighbor of y. The numbers dD(x) = d (x) = |N+(x)| and dD(x) = d−(x) = |N−(x)| are called the outdegree and indegree of x, respectively. For a vertex set X of D, we define D[X ] as the subdigrah induced by X . If we speak of a cycle, then we mean a directed cycle, and a cycle of length n is called an n-cycle. If we replace in a digraph D every arc xy by yx, then we call the resulting digraph the converse of D, denoted by D−1. There are several measures of how much a digraph differs from being regular. In [14], Yeo defines the global irregularity of a digraph D by ig(D) = max x∈V (D) {d+(x), d−(x)} − min y∈V (D) {d+(y), d−(y)}. If ig(D) = 0, then D is regular and if ig(D) ≤ 1, then D is called almost regular. A c-partite or multipartite tournament is an orientation of a complete c-partite graph. A tournament is a c-partite tournament with exactly c vertices. If V1, V2, . . . , Vc are the partite sets of a c-partite tournament D and the vertex x of D belongs to the partite set Vi, then we define V (x) = Vi. If D is a c-partite tournament with the partite sets V1, V2, . . . , Vc such that |V1| ≤ |V2| ≤ . . . ≤ |Vc|, then |Vc| = α(D) is the independence number of D, and we define γ(D) = |V1|. It is very easy to see that every arc of a regular tournament belongs to a 3-cycle. The next example shows that this is not valid for regular multipartite tournaments in general. Example 1.1 Let C,C ′, and C ′′ be three induced cycles of length 4 such that C → C ′ → C ′′ → C. The resulting 6-partite tournament D1 is 5-regular, but no arc of the three cycles C,C ′, C ′′ is contained in a 3-cycle. Let H,H1, and H2 be three copies of D1 such that H → H1 → H2 → H. The resulting 18-partite tournament is 17-regular, but no arc of the cycles corresponding to the cycles C,C ′, and C ′′ is contained in a 3-cycle. If we continue this process, we arrive at regular c-partite tournaments with arbitrary large c which contain arcs that do not belong to any 3-cycle. In 1998, Guo [3] proved the following generalization of Alspach’s classical result [1] that every regular tournament is arc pancyclic. Theorem 1.2 (Guo [3]) Let D be a regular c-partite tournament with c ≥ 3. If every arc of D is contained in a 3-cycle, then every arc of D is contained in an n-cycle for each n ∈ {4, 5, . . . , c}. Now, the aim was to carry this result forward to almost regular multipartite tournaments. To reach this, Volkmann [10], [12] started with the following theorems. Theorem 1.3 (Volkmann [12]) Let D be an almost regular multipartite tournament with c partite sets. If c ≥ 8, then every arc of D is contained in a 4-cycle.