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

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DOI

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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     title = {Complete {Diagonals} of {Latin} {Squares}},
     journal = {Canadian mathematical bulletin},
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     year = {1979},
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     doi = {10.4153/CMB-1979-062-3},
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