Designs, Codes and Cryptography

, Volume 52, Issue 1, pp 93–105 | Cite as

Indivisible plexes in latin squares

  • Darryn Bryant
  • Judith EganEmail author
  • Barbara Maenhaut
  • Ian M. Wanless


A k-plex is a selection of kn entries of a latin square of order n in which each row, column and symbol is represented precisely k times. A transversal of a latin square corresponds to the case k = 1. A k-plex is said to be indivisible if it contains no c-plex for any 0 < c < k. We prove that if n = 2km for integers k ≥ 2 and m ≥ 1 then there exists a latin square of order n composed of 2m disjoint indivisible k-plexes. Also, for positive integers k and n satisfying n = 3k, n = 4k or n ≥ 5k, we construct a latin square of order n containing an indivisible k-plex.


Latin square Transversal Plex Orthogonal partition 

Mathematics Subject Classifications (2000)

05B15 62K99 


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  1. 1.
    Cavenagh N.J., Donovan D.M., Yazici E.S.: Minimal homogeneous latin trades. Discrete Math. 306, 2047–2055 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Colbourn C.J., Rosa A.: Triple Systems. Clarendon Press, Oxford (1999)zbMATHGoogle Scholar
  3. 3.
    Dénes J., Keedwell A.D.: Latin squares and their applications. Akadémiai Kiadó, Budapest (1974)zbMATHGoogle Scholar
  4. 4.
    Egan J., Wanless I.M.: Latin squares with no small odd plexes. J. Comb. Des. 16, 477–492 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Egan J., Wanless I.M.: Indivisible partitions of latin squares (preprint).Google Scholar
  6. 6.
    Hall P.: On representatives of subsets. J. London Math. Soc. 10, 26–30 (1935)zbMATHCrossRefGoogle Scholar
  7. 7.
    Wanless I.M.: A generalisation of transversals for latin squares. Electron. J. Comb. 9, R12 (2002)MathSciNetGoogle Scholar
  8. 8.
    Wanless I.M.: Diagonally cyclic latin squares. Eur. J. Comb. 25, 393–413 (2004)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Darryn Bryant
    • 1
  • Judith Egan
    • 2
    Email author
  • Barbara Maenhaut
    • 1
  • Ian M. Wanless
    • 2
  1. 1.Department of MathematicsUniversity of QueenslandQLDAustralia
  2. 2.School of Mathematical SciencesMonash UniversityVICAustralia

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