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Complete Diagonals of Latin Squares

Published online by Cambridge University Press:  20 November 2018

Gerard J. Chang
Gerard J. Chang*
Affiliation:
184 Cornell Quarters, Ithaca, N.Y. 14850 U.S.A.
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Abstract

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J. Marica and J. Schönhein [4], using a theorem of M. Hall, Jr. [3], see below, proved that if any n − 1 arbitrarily chosen elements of the diagonal of an n × n array are prescribed, it is possible to complete the array to form an n × n latin square. This result answers affirmatively a special case of a conjecture of T. Evans [2], to the effect that an n × n incomplete latin square with at most n − 1 places occupied can be completed to an n × n latin square. When the complete diagonal is prescribed, it is easy to see that a counterexample is provided by the case that one letter appears n − 1 times on the diagonal and a second letter appears once. In the present paper, we prove that except in this case the completion to a full latin square is always possible. Completion to a symmetric latin square is also discussed.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 22 , Issue 4 , 01 December 1979 , pp. 477 - 481
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Dẽnes, J. and Keedwell, A. D., Latin squares and their applications, Academic Press, New York and London, 1974.Google Scholar
2. Evans, T., Embedding incomplete latin squares, Amer. Math. Monthly vol. 67 (1960) pp. 958-961.Google Scholar
3. Hall, M. Jr, A combinatorial problem on abelian groups, Proc. Amer. Math. Soc. vol. 3 (1952) pp. 584-587.Google Scholar
4. Marica, J. and Schönheim, J., Incomplete diagonals of latin squares, Canad. Math. Bull. vol. 12 (1969) pp. 235.Google Scholar