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Complementary Cycles in Irregular Multipartite Tournaments

2016
*
Mathematical Problems in Engineering
*

A

doi:10.1155/2016/5384190
fatcat:tgane664lzdevjpwg6fc44u6he
*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. ...##
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Vertex deletion and cycles in multipartite tournaments

1999
*
Discrete Mathematics
*

A

doi:10.1016/s0012-365x(98)00277-5
fatcat:idfpm3jc4zebdi775cmwqdodpu
*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. ...##
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Vertex deletion and cycles in multipartite tournaments

1999
*
Discrete Mathematics
*

A

doi:10.1016/s0012-365x(99)90145-0
fatcat:mar4ogte4fd23ei53pi6urhd5m
*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. ...##
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Cycles in multipartite tournaments: results and problems

2002
*
Discrete Mathematics
*

Many results about

doi:10.1016/s0012-365x(01)00419-8
fatcat:rg3yfj4nazaj5afljiv4hbyrvq
*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. ...##
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On cycles through two arcs in strong multipartite tournaments
[article]

2010
*
arXiv
*
pre-print

A

arXiv:1006.0902v1
fatcat:n6vtmgdktzdnbftb7uwqwytx2a
*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*. ...##
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Weakly Complementary Cycles in 3-Connected Multipartite Tournaments

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*. ...

##
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On cycles through a given vertex in multipartite tournaments

1997
*
Discrete Mathematics
*

An n-partite

doi:10.1016/s0012-365x(96)00048-9
fatcat:kquhss3rcrholh4zi3slbuzkye
*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. ...##
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Cycles Containing a Given Arc in Regular Multipartite Tournaments

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. ...

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

2009
*
Discrete Mathematics
*

Let D be a

doi:10.1016/j.disc.2008.08.022
fatcat:5dtgigzwy5fkjjliwphoxpllaa
*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*. ...##
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On Cycles Containing a Given Arc in Regular Multipartite Tournaments

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*...

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

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. ...

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

2011
*
Applied Mathematics Letters
*

Volkmann, A remark on

doi:10.1016/j.aml.2010.10.022
fatcat:wryv4x6ekvecnccyoio66lh2bq
*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. ...##
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Cycles through a given arc and certain partite sets in almost regular multipartite tournaments

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. ...

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

2007
*
Journal of the Korean Mathematical Society
*

Here we will examine the existence of

doi:10.4134/jkms.2007.44.3.683
fatcat:wpxukygrijfo5lq4ax6ghckomu
*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] . ...##
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Paths and cycles containing given arcs, in close to regular multipartite tournaments

2007
*
Journal of combinatorial theory. Series B (Print)
*

Finally we give a lower bound on the number of Hamilton

doi:10.1016/j.jctb.2007.02.004
fatcat:u3kqujjbtvawjdiv6qsovz6s64
*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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