Embedding partial Latin squares with prescribed diagonal

For more than 30 years, one of the most difficult embedding problems concerning Latin squares has been that of completing partial idempotent Latin squares and analogues of it. There have been many results in the area, including the solution of some hard problems. One problem which is still unsolved is that of finding a necessary and sufficient condition for a partial Latin square of side r to be embeddable in a Latin square of side n with given symbols on the diagonal. This project concentrates on the case where no symbol is prescribed exactly once on the diagonal.