Stefan Winzen

Weakly Complementary Cycles in 3-Connected Multipartite Tournaments

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. A digraph D is called cycle complementary if there exist two vertex disjoint cycles C1 and C2 such that V (D) = V (C1) ( V (C2), and a multipartite tournament D is called weakly cycle complementary if there exist two vertex disjoint cycles C1 and C2 such that V (C1) ( V (C2) contains vertices of all partite sets of D. The problem of complementary cycles in 2-connected tournaments was completely solved by Reid (4) in 1985 and Z. Song (5) in 1993. They proved that every 2-connected tournament T on at least 8 vertices has complementary cycles of length t and jV (T)j i t for all 3 • tjV (T)j=2. Recently, Volkmann (8) proved that each regular multipartite tournament D of order jV (D)j ‚ 8 is cycle complementary. In this article, we analyze multipartite tournaments that are weakly cycle complementary. Especially, we will characterize all 3-connected c-partite tournaments with c ‚ 3 that are weakly cycle complementary. 1. Terminology In this paper all digraphs are flnite 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 ; Y denotes the fact that there is no arc leading from Y to X. 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 i D (x) = N i (x) is the set of vertices dominating x. Therefore, if the arc xy 2 E(D) exists, then y is an outer neighbor of x and x is an inner neighbor of y. The numbers d + (x) = d + (x) = jN + (x)j and d i (x) = d i (x) = jN i (x)j are called the outdegree and the indegree of x, respectively. Furthermore, the numbers - + D = - + = minfd + (x)jx 2 V (D)g and - i D = - i = minfd i (x)jx 2 V (D)g are the minimum outdegree and the minimum

Almost regular multipartite tournaments containing a Hamiltonian path through a given arc

Abstract 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. 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 called almost regular. Recently, Volkmann and Yeo have proved that every arc of a regular multipartite tournament is contained in a directed Hamiltonian path. If c⩾4, then this result remains true for almost all c-partite tournaments D of a given constant irregularity ig(D). For the case that ig(D)=1 we will give a more detailed analysis. If c=3, then there exist infinite families of such digraphs, which have an arc that is not contained in a directed Hamiltonian path of D. Nevertheless, we will present an interesting sufficient condition for an almost regular 3-partite tournament D with the property that a given arc is contained in a Hamiltonian path of D.

Complementary cycles in regular multipartite tournaments, where one cycle has length five

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 1999, Yea conjectured that each regular c-partite tournament D with and contains a pair of vertex disjoint directed cycles of lengths 4 and . An example will demonstrate that Yeos conjecture is not true in general for regular 4-partite tournaments with two vertices in each partite. However, in all other cases we shall confirm this conjecture in affirmative.

On the connectivity of close to 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) and the local irregularity of a digraph D is il(D) = max |d+(x) - d- (x)| over all vertices x of D. Clearly, il(D) ≤ ig(D). 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. Let V1, V2,.....,Vc be the partite sets of a c-partite tournament such that |V1| ≤ |V2| ≤....≤ |Vc|. In 1998, Yeo proved K(D)≤ ⌈ |V(D)| - |Vc| - 2il(D)/3⌉ for each c-partite tournament D, where k(D) is the connectivity of D. Using Yeos proof, we will present the structure of those multipartite tournaments, which fulfill the last inequality with equality. These investigations yield the better bound K(D)≥ ⌈ |V(D)| - |Vc| - 2il(D + 1)/3 ⌈ in the case that |Vc| is odd. Especially, we obtain a 1980 result by Thomassen for tournaments of arbitrary (global) irregularity. Furthermore, we will give a shorter proof of the recent result of Volkmann that K(D)≥ ⌈ |V(D)| - |Vc| + 1 /3⌉ for all regular multipartite tournaments with exception of a well-determined family of regular (3q + 1)-partite tournaments. Finally we will characterize all almost regular tournaments with this property.

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 A 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.

Cycles through a given arc and certain partite sets in almost regular multipartite tournaments

Abstract 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 i g ( D )=max{ d + ( x ), d − ( x )}−min{ d + ( y ), d − ( y )} over all vertices x and y of D (including x = y ). If i g ( D )=0, then D is regular and if i g ( D )⩽1, then D is almost regular. A c -partite tournament is an orientation of a complete c -partite graph. In 1998, Guo and Kwak showed that, if D is a regular c -partite tournament with c ⩾4, then every arc of D is in a directed cycle, which contains vertices from exactly m partite sets for all m ∈{4,5,…, c }. In this paper we shall extend this theorem to almost regular c -partite tournaments, which have at least two vertices in each partite set. An example will show that there are almost regular c -partite tournaments with arbitrary large c such that not all arcs are in directed cycles through exactly 3 partite sets. Another example will show that the result is not valid for the case that c =4 and there is only one vertex in a partite set.