Journal of Applied Mathematics and Computing

, Volume 40, Issue 1–2, pp 249–259 | Cite as

The size of 3-compatible, weakly compatible split systems

  • Stefan Grünewald
  • Jack H. Koolen
  • Vincent Moulton
  • Taoyang Wu
Applicable mathematics in biological


A split system on a finite set X is a set of bipartitions of X. Weakly compatible and k-compatible (k≥1) split systems are split systems which satisfy special restrictions on all subsets of a certain fixed size. They arise in various areas of applied mathematics such as phylogenetics and multi-commodity flow theory. In this note, we show that the number of splits in a 3-compatible, weakly compatible split system on a set X of size n is linear in n.


Phylogenetic combinatorics Extremal combinatorics of finite sets Split systems Compatibility Weak compatibility 

Mathematics Subject Classification (2000)

05D05 03E05 92B05 



TW and VM were supported by the Engineering and Physical Sciences Research Council [grant number EP/D068800/1]. VM thanks the Royal Society for enabling him to visit TW and JK in Singapore. TW was also partially supported by the Singapore MOE grant R-146-000-134-112. JK was partially supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (grant number 2010-0008138).


  1. 1.
    Bandelt, H.-J., Dress, A.: A canonical decomposition theory for metrics on a finite set. Adv. Math. 92, 47–105 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bandelt, H.-J., Dress, A.: Split decomposition: A new and useful approach to phylogenetic analysis of distance data. Mol. Phylogenet. Evol. 1, 242–252 (1992) CrossRefGoogle Scholar
  3. 3.
    Capoyleas, V., Pach, J.: A Turán-type theorem on chords of a convex polygon. J. Comb. Theory, Ser. B 56, 9–15 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Dress, A., Klucznik, M., Koolen, J., Moulton, V.: A note on extremal combinatorics of cyclic split system. Sémin. Lothar. Comb. 47, B47b (2001) MathSciNetGoogle Scholar
  5. 5.
    Dress, A., Koolen, J., Moulton, V.: On line arrangements in the hyperbolic plane. Eur. J. Comb. 23, 549–557 (2002). (See also Addendum, Eur. J. Comb. 24, 347 (2003)) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dress, A., Koolen, J., Moulton, V.: 4n-10. Ann. Comb. 8, 463–471 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Fleiner, T.: The size of 3-cross-free families. Combinatorica 21, 445–448 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Grünewald, S., Koolen, J., Lee, W.S.: Quartets in maximal weakly compatible split systems. Appl. Math. Lett. 22(6), 1604–1608 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Huson, D., Bryant, D.: Application of phylogenetic networks in evolutionary studies. Mol. Biol. Evol. 23(2), 254–267 (2006) CrossRefGoogle Scholar
  10. 10.
    Karzanov, A.: Combinatorial methods to solve cut-determined multiflow problems. In: Karzanov, A. (ed.) Combinatorial Methods for Flow Problems, no. 3, pp. 6–69. Vsesoyuz. Nauchno-Issled. Inst. Sistem. Issled., Moscow (1979) (in Russian) Google Scholar
  11. 11.
    Karzanov, A.V., Lomonosov, M.V.: Flow systems in undirected networks. In: Larichev, O.I. (ed.) Mathematical Programming, pp. 59–66. Institute for System Studies, Moscow (1978) (in Russian) Google Scholar
  12. 12.
    Nakamigawa, T.: A generalization of diagonal flips in a convex polygon. Theor. Comput. Sci. 235, 271–282 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Pevzner, P.: Non-3-crossing families and multicommodity flows. Am. Math. Soc. Transl. Ser. 2 158, 201–206 (1994). (Translated from: Pevzner, P., Linearity of the cardinality of 3-cross free sets. In: Fridman, A. (ed.) Problems of Discrete Optimization and Methods for Their Solution, pp. 136–142, Moscow (1987) (in Russian)). MathSciNetGoogle Scholar
  14. 14.
    Pilaud, V., Santos, F.: Multitriangulations as complexes of star polygons. Discrete Comput. Geom. 41, 284–317 (2009) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2012

Authors and Affiliations

  • Stefan Grünewald
    • 1
  • Jack H. Koolen
    • 2
  • Vincent Moulton
    • 3
  • Taoyang Wu
    • 3
    • 4
  1. 1.CAS-MPG Partner Institute for Computational Biology (PICB), Key Laboratory of Computational BiologyShanghai Institutes for Biological SciencesShanghaiP.R. China
  2. 2.Department of MathematicsPOSTECHPohangSouth Korea
  3. 3.School of Computing SciencesUniversity of East AngliaNorwichUK
  4. 4.Department of MathematicsNational University of SingaporeSingaporeSingapore

Personalised recommendations