Multipartite tournaments: A survey

Self Cite

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 byIf i g (D) = 0, then D is regular and if i g (D) ≤ 1, then D is called almost regular. A c-partite tournament is an orientation of a complete c-partite graph. Recently, Volkmann and Winzen [L. Volkmann, S. Winzen, Almost regular c-partite tournaments contain a strong subtournament of order c when c ≥ 5, Discrete Math. (2007), 10.1016/j.disc.2006.10.019] showed that every almost regular c-partite tournament D with c ≥ 5 contains a strongly connected subtournament of order p for every p ∈ {3, 4, . . . , c}. In this paper for the class of regular multipartite tournaments we will consider the more difficult question for the existence of strong subtournaments containing a given vertex. We will prove that each vertex of a regular multipartite tournament D with c ≥ 7 partite sets is contained in a strong subtournament of order p for every p ∈ {3, 4, . . . , c − 4}.

Section: Terminology and Introductionmentioning

confidence: 99%

Strong subtournaments containing a given vertex in regular multipartite tournaments

Volkmann

Winzen

2008

Self Cite

“…. , c} in general [10]. However, Zhou et al [12] proved that every vertex of a regular c-partite tournament with at least four partite sets (c ≥ 4) is contained in a cycle of length m for each m ∈ {3, .…”

Section: Introductionmentioning

confidence: 98%

“…A very recent survey on this topic [10] appeared with several interesting open problems. For instance, the study of cycles whose length does not exceed the number of partite sets leads to various extensions and generalizations of classic results on tournaments.…”

Section: Introductionmentioning

confidence: 99%

The number of C3⃗-free vertices on 3-partite tournaments

Figueroa

Llano

Zuazua

2010

a b s t r a c tLet T be a 3-partite tournament. We say that a vertex v is − → C 3 -free if v does not lie on any directed triangle of T . Let F 3 (T ) be the set of the − → C 3 -free vertices in a 3-partite tournament and f 3 (T ) its cardinality. In this paper we prove that if T is a regular 3-partite tournament, then F 3 (T ) must be contained in one of the partite sets of T . It is also shown that for every regular 3-partite tournament, f 3 (T ) does not exceed n 9 , where n is the order of T . On the other hand, we give an infinite family of strongly connected tournaments having n − 4 − → C 3 -free vertices. Finally we prove that for every c ≥ 3 there exists an infinite family of strongly connected c-partite tournaments, D c (T ), with n − c − 1 − → C 3 -free vertices.

“…The structure of cycles in multipartite tournaments has been extensively studied, see for example [6,5]. In 1998, Zhou et al [8] has proved that, if T is a regular c-partite tournament with c ≥ 4, then T does not have − → C 3 -free vertices.…”

Section: Introductionmentioning

confidence: 99%

On the vertices of a 3-partite tournament not in triangles

Figueroa

Olsen²,

Zuazua

2015

a b s t r a c tLet T be a 3-partite tournament and F 3 (T ) be the set of vertices of T not in triangles. We prove that, if the global irregularity of T , i g (T ), is one and |F 3 (T )| > 3, then F 3 (T ) must be contained in one of the partite sets of T and |F 3 (T )| ≤  k+1 4  + 1, which implies |F 3 (T )| ≤  n+5 12  + 1, where k is the size of the largest partite set and n the number of vertices of T .Moreover, we give some upper bounds on the number, as well as results on the structure of said vertices within the digraph, depending on its global irregularity.