Completing Latin squares

  • Martin Aigner
  • Günter M. Ziegler


Some of the oldest combinatorial objects, whose study apparently goes back to ancient times, are the Latin squares. To obtain a Latin square, one has to fill the n 2 cells of an (n × n)-square array with the numbers 1, 2,..., n so that that every number appears exactly once in every row and in every column. In other words, the rows and columns each represent permutations of the set {1,..., n}. Let us call n the order of the Latin square.


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    T. Evans: Embedding incomplete Latin squares, Amer. Math. Monthly 67 (1960), 958–961.MathSciNetCrossRefGoogle Scholar
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    C. C. Lindner: On completing Latin rectangles, Canadian Math. Bulletin 13 (1970), 65–68.MathSciNetzbMATHCrossRefGoogle Scholar
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    H. J. Ryser: A combinatorial theorem with an application to Latin rectangles, Proc. Amer. Math. Soc. 2 (1951), 550–552.MathSciNetzbMATHCrossRefGoogle Scholar
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    B. Smetaniuk: A new construction on Latin squares I: A proof of the Evans conjecture, Ars Combinatoria 11 (1981), 155–172.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Martin Aigner
    • 1
  • Günter M. Ziegler
    • 2
  1. 1.Institut für Mathematik II (WE2)Freie Universität BerlinBerlinGermany
  2. 2.Institut für Mathematik, MA 6-2Technische Universität BerlinBerlinGermany

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