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CYCLES THROUGH A GIVEN SET OF VERTICES IN REGULAR MULTIPARTITE TOURNAMENTS

  • Volkmann, Lutz ;
  • Winzen, Stefan
  • Published : 2007.05.31

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. In a recent article, the authors proved that a regular c-partite tournament with $r{\geq}2$ 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 $X{\subseteq}V(D)$ be an arbitrary set with exactly 2 vertices of each partite set. For all $c{\geq}4$ we will determine the minimal value g(c) such that D-X is Hamiltonian for every regular multipartite tournament with $r{\geq}g(c)$.

Keywords

multipartite tournaments;regular multipartite tournaments;cycles through given set of vertices

References

  1. J. Bang-Jensen and G. Gutin, Digraphs, Theory, algorithms and applications. Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London, 2001
  2. Y. Guo, Semi complete Multipartite Digraphs: A Generalization of Tournaments, Habilitation thesis, RWTH Aachen (1998), 102 pp
  3. G. Gutin, Cycles and paths in semicomplete multipartite digraphs, theorems, and algorithms: a survey, J. Graph Theory 19 (1995), no. 4, 481-505 https://doi.org/10.1002/jgt.3190190405
  4. J. W. Moon, On subtournaments of a tournament, Canad. Math. Bull. 9 (1966), 297-301 https://doi.org/10.4153/CMB-1966-038-7
  5. O. Ore, Theory of gmphs, American Mathematical Society Colloquium Publications, Vol. 38, American Mathematical Society, 1962
  6. J. Stella, L. Volkmann, and S. Winzen, How close to regular must a multipartite tournament be to secure a given path covering number?, Ars Combinatoria, to appea
  7. P. Turan, Eine Extremalaufgabe aus der Gmphentheorie, Mat. Fiz. Lapok 48 (1941), 436-452
  8. L. Volkmann, Strong subtournaments of multipartite tournaments, Australas. J. Combin. 20 (1999), 189-196
  9. L. Volkmann, Cycles in multipartite tournaments: results and problems, Discrete Math. 245 (2002), no. 1-3, 19-53 https://doi.org/10.1016/S0012-365X(01)00419-8
  10. L. Volkmann and S. Winzen, Cycles with a given number of vertices from each partite set in regular multipartite tournaments, Czechoslovak Math. J. 56 (131) (2006), no. 3, 827-843 https://doi.org/10.1007/s10587-006-0059-8
  11. L. Volkmann and S. Winzen, On the connectivity of close to regular multipartite tournaments, Discrete Appl. Math. 154 (2006), no. 9, 1437-1452 https://doi.org/10.1016/j.dam.2004.09.022
  12. S. Winzen, Close to Regular Multipartite Tournaments, Ph. D. thesis, RWTH Aachen, 2004
  13. A. Yeo, One-diregular subgmphs in semicomplete multipartite digmphs, J. Graph Theory 24 (1997), no. 2, 175-185 https://doi.org/10.1002/(SICI)1097-0118(199702)24:2<175::AID-JGT5>3.0.CO;2-N
  14. A. Yeo, Semicomplete Multipartite Digmphs, Ph. D. thesis, Odense University, 1998
  15. A. Yeo, How close to regular must a semicomplete multipartite digmph be to secure Hamiltonicity?, Graphs Combin. 15 (1999), no. 4, 481-493 https://doi.org/10.1007/s003730050080
  16. G. Gutin, Note on the path covering number of a semicomplete multipartite digraph, J. Combin. Math. Combin. Comput. 32 (2000), 231-237
  17. L. Volkmann, F'undamente der Gmphentheorie, Springer Lehrbuch Mathematik, Springer-Verlag, Vienna, 1996
  18. L. Volkmann and S. Winzen, Almost regular c-partite tournaments contain a strong subtournaments of order c when c${ge}$ 5, Discrete Math., to appear

Cited by

  1. Multipartite tournaments: A survey vol.307, pp.24, 2007, https://doi.org/10.1016/j.disc.2007.03.053