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On Completing Latin Rectangles

Published online by Cambridge University Press:  20 November 2018

Charles C. Lindner
Charles C. Lindner*
Affiliation:
Auburn University, Auburn, Alabama
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By an (incomplete) r × s latin rectangle is meant an r × s array such that (in some subset of the rs cells of the array) each of the cells is occupied by an integer from the set 1, 2, …, s and such that no integer from the set 1,2, …, s occurs in any row or column more than once. This definition requires that r≦s. If r=s we will replace the word rectangle by square. It is easy to see that for any n≧2 there is an incomplete n × 2n latin rectangle with 2n cells occupied which cannot be completed to a n × 2n latin rectangle. In this paper we prove the following theorem.

Theorem 1. An incomplete n × 2n latin rectangle with 2n—1 cells occupied can be completed to a n × 2n latin rectangle.

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 13 , Issue 1 , March 1970, pp. 65 - 68
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Evans, T., Embedding incomplete latin squares, Amer. Math. Monthly 67 (1960), 958-961.Google Scholar
2. Hall, M., An existence theorem for latin squares, Bull. Amer. Math. Soc. (1945), 387-388.Google Scholar
3. Hall, P., On representatives of subsets, J. London Math. Soc. 10 (1935), 26-32.Google Scholar
4. Marica, J. and Schonheim, J., Incomplete diagonals of latin squares, Canad. Math. Bull. 12 (1969), 235.Google Scholar
5. Ryser, H. J., Combinatorial Mathematics, The Carus Mathematical Monographs, No. XIV, Math. Assoc. Amer. (1963).Google Scholar