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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
Article

Abstract

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.

Keywords

Latin square Transversal Plex Orthogonal partition 

Mathematics Subject Classifications (2000)

05B15 62K99 

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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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