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### Complementary Cycles in Irregular Multipartite Tournaments

Zhihong He, Xiaoying Wang, Caiming Zhang
2016 Mathematical Problems in Engineering
A tournament is a directed graph obtained by assigning a direction for each edge in an undirected complete graph.  ...  In this paper, we prove that ifD-V(C3)has no cycle factor, thenDcontains a pair of disjoint cycles of length3and|V(D)|-3, unlessDis isomorphic toT7,D4,2,D4,2⁎, orD3,2.  ...  Let be a multipartite tournament. If ( ) ≥ ( ) + 1, then is cycle complementary, unless is a member of a finite family of multipartite tournaments. In 2010, Li et al.  ...

### Vertex deletion and cycles in multipartite tournaments

M Tewes
1999 Discrete Mathematics
A tournament is an orientation of a complete graph and a multipartite tournament is an orientation of a complete multipartite graph.  ...  Therefore, a tournament is a k-partite tournament with exactly k vertices, From the well-known theorem of Moon that every vertex of a strong tournament T is contained in a directed cycle of length m for  ...  Cycles in multipartite tournaments Let D be a strongly connected k-partite tournament with k/>3.  ...

### Vertex deletion and cycles in multipartite tournaments

Meike Tewes, Lutz Volkmann
1999 Discrete Mathematics
A tournament is an orientation of a complete graph and a multipartite tournament is an orientation of a complete multipartite graph.  ...  Therefore, a tournament is a k-partite tournament with exactly k vertices, From the well-known theorem of Moon that every vertex of a strong tournament T is contained in a directed cycle of length m for  ...  Cycles in multipartite tournaments Let D be a strongly connected k-partite tournament with k/>3.  ...

### Cycles in multipartite tournaments: results and problems

Lutz Volkmann
2002 Discrete Mathematics
Many results about cycles in tournaments are known, but closely related problems involving cycles in multipartite tournaments have received little attention until recently.  ...  A tournament is an orientation of a complete graph, and in general a multipartite tournament is an orientation of a complete n-partite graph.  ...  Hammer for the wonderful idea and for his kind encouragement to publish this paper in the "Perspectives" section.  ...

### On cycles through two arcs in strong multipartite tournaments [article]

Alexandru I. Tomescu
2010 arXiv   pre-print
A multipartite tournament is an orientation of a complete c-partite graph. In [L. Volkmann, A remark on cycles through an arc in strongly connected multipartite tournaments, Appl. Math.  ...  Lett. 20 (2007) 1148--1150], Volkmann proved that a strongly connected c-partite tournament with c > 3 contains an arc that belongs to a directed cycle of length m for every m ∈{3, 4, ..., c}.  ...  In [3] , Volkmann showed that a similar result holds for the case of strong multipartite tournaments.  ...

### Weakly Complementary Cycles in 3-Connected Multipartite Tournaments

Lutz Volkmann, Stefan Winzen
2008 Kyungpook Mathematical Journal
In this article, we analyze multipartite tournaments that are weakly cycle complementary.  ...  The problem of complementary cycles in 2-connected tournaments was completely solved by Reid [4] in 1985 and Z. Song [5] in 1993.  ...  In 1993, Song [5] extended this result. The problem of complementary cycles in multipartite tournaments is much more difficult to analyze than in tournaments.  ...

### On cycles through a given vertex in multipartite tournaments

Yubao Guo, Axel Pinkernell, Lutz Volkmann
1997 Discrete Mathematics
An n-partite tournament is an orientation of a complete n-partite graph, and an m-cycle is a directed cycle of length m.  ...  If D is a strongly connected n-partite tournament with the partite sets V1, V2 .... , V" and v an arbitrary vertex of D, then we shall prove the following statements. • The vertex v is contained in a longest  ...  Introduction An n-partite or multipartite tournament is an orientation of a complete n-partite graph. A tournament is an n-partite tournament with exactly n vertices.  ...

### Cycles Containing a Given Arc in Regular Multipartite Tournaments

Guo-fei Zhou, Ke-min Zhang
2002 Acta Mathematicae Applicatae Sinica (English Series)
In this paper we prove that if T is a regular n-partite tournament with n≥6, then each arc of T lies on a k-cycle for k=4,5,···,n.  ...  A k-outpath of an arc xy in a multipartite tournament is a directed path with length k starting from xy such that x does not dominate the end vertex of the directed path.  ...  Let T be a multipartite tournament and x ∈ V (T ), we use V (x) to denote the partite set of T to which x belongs.  ...

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

Zhihong He, Torsten Korneffel, Dirk Meierling, Lutz Volkmann, Stefan Winzen
2009 Discrete Mathematics
Let D be a multipartite tournament. If κ(D) ≥ α(D) + 1, then D is cycle complementary, unless D is a member of a finite family of multipartite tournaments.  ...  The problem of complementary cycles in tournaments was almost completely solved by Reid [4] in 1985 and by Z. Song [5] in 1993.  ...  Let D be a multipartite tournament having a cycle-factor but no Hamiltonian cycle.  ...

### On Cycles Containing a Given Arc in Regular Multipartite Tournaments

Lin Qiang Pan, Ke Min Zhang
2004 Acta Mathematica Sinica. English series
In this paper we prove that if T is a regular n-partite tournament with n ≥ 4, then each arc of T lies on a cycle whose vertices are from exactly k partite sets for k = 4, 5, . . . , n.  ...  Our result, in a sense, generalizes a theorem due to Alspach.  ...  Many thanks are also given to the anonymous referees for their useful comments, the correction of the proof of Lemma 1, and pointing out the existence of Theorem E and its relation with the main theorem in  ...

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

L VOLKMANN
2004 Discrete Mathematics
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;  ...  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.  ...  Let D be a c-partite tournament with ig(D) 6 i and r vertices in each partite set.  ...

### Cycles through arcs in multipartite tournaments and a conjecture of Volkmann

Hongwei Li, Shengjia Li, Yubao Guo, Qiaoping Guo
2011 Applied Mathematics Letters
Volkmann, A remark on cycles through an arc in strongly connected multipartite tournaments, Appl. Math.  ...  Lett. 20 (2007) 1148-1150] conjectured that a strong c-partite tournament with c ≥ 3 contains three arcs that belong to a cycle of length m for each m ∈ {3, 4, . . . , c}.  ...  Introduction and preliminaries A multipartite tournament or c-partite tournament is an orientation of a complete c-partite graph. A tournament is a c-partite tournament with exactly c vertices.  ...

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

Lutz Volkmann, Stefan Winzen
2004 Discrete Mathematics
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;  ...  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.  ...  Let D be a c-partite tournament with ig(D) 6 i and r vertices in each partite set.  ...

### CYCLES THROUGH A GIVEN SET OF VERTICES IN REGULAR MULTIPARTITE TOURNAMENTS

Lutz Volkmann, Stefan Winzen
2007 Journal of the Korean Mathematical Society
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.  ...  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.  ...  There is an extensive literature on cycles in multipartite tournaments, see e.g., Bang-Jensen and Gutin [1] , Guo [2] , Gutin [3] , Volkmann [11] , Winzen [15] and Yeo [17] .  ...

### Paths and cycles containing given arcs, in close to regular multipartite tournaments

Anders Yeo
2007 Journal of combinatorial theory. Series B (Print)
Finally we give a lower bound on the number of Hamilton cycles in a c-partite tournament with c 4.  ...  Sufficient conditions are furthermore given for when a c-partite tournament with c 4 has a Hamilton cycle containing a given path or a set of given arcs.  ...  Introduction and terminology A c-partite or multipartite tournament (MT) is an orientation of a complete c-partite graph. A tournament is a c-partite tournament with exactly c vertices.  ...
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