Complete diagonals of Latin squares

Gerard J. Chang
1979 Canadian mathematical bulletin  
J. Marica and J. Schônhein [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 nxn array are prescribed, it is possible to complete the array to form an n x n latin square. This result answers affirmatively a special case of a conjecture of T. Evans [2] , to the effect that an n x n incomplete latin square with at most n-1 places occupied can be completed to an nxn latin square. When the complete diagonal is prescribed,
more » ... 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.
doi:10.4153/cmb-1979-062-3 fatcat:m4sad7lhdnbwnkzgwd2kpvx4ee