Cycles through a given Set of Vertices in Regular Multipartite Tournaments

A tournament is an orientation of a complete graph, and in general a multipartite or c-partite tournament is an orientation of a complete c-partite graph. In a recent article, the authors proved that a regular c-partite tournament with r ≥ 2 vertices in each partite set contains a cycle with exactly r− 1 vertices from each partite set, with exception of the case that c = 4 and r = 2. Here we will examine the existence of cycles with r−2 vertices from each partite set in regular multipartite tournaments where the r−2 vertices are chosen arbitrarily. Let D be a regular c-partite tournament and let X ⊆ V (D) be an arbitrary set with exactly 2 vertices of each partite set. For all c ≥ 4 we will determine the minimal value g(c) such that D−X is Hamiltonian for every regular multipartite tournament with r ≥ g(c). 1. Terminology and introduction In this paper all digraphs are finite without loops and multiple arcs. The vertex set and the 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 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 Ã 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 out-neighborhood N D (x) = N (x) of a vertex x is the set of vertices dominated by x and the in-neighborhood N− D (x) = N −(x) is the set of vertices dominating x. Therefore, if xy ∈ E(D), then y is an outer neighbor of x and x is an inner neighbor of y. The numbers d+D(x) = d(x) = |N+(x)| and dD(x) = d−(x) = |N−(x)| are called the outdegree and the indegree of x, respectively. Furthermore, the numbers δ D = δ + = Received December 19, 2005. 2000 Mathematics Subject Classification. 05C20.