Geodesic
Complete Diagonals of Latin Squares
Canadian mathematical bulletin, Tome 22 (1979) no. 4, pp. 477-481

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Chang, Gerard J. Complete Diagonals of Latin Squares. Canadian mathematical bulletin, Tome 22 (1979) no. 4, pp. 477-481. doi: 10.4153/CMB-1979-062-3
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[1] 1. Dẽnes, J. and Keedwell, A. D., Latin squares and their applications, Academic Press, New York and London, 1974. Google Scholar

[2] 2. Evans, T., Embedding incomplete latin squares, Amer. Math. Monthly vol. 67 (1960) pp. 958-961. Google Scholar

[3] 3. Hall, M. Jr, A combinatorial problem on abelian groups, Proc. Amer. Math. Soc. vol. 3 (1952) pp. 584-587. Google Scholar

[4] 4. Marica, J. and Schönheim, J., Incomplete diagonals of latin squares, Canad. Math. Bull. vol. 12 (1969) pp. 235. Google Scholar

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