Completing partial latin squares

Abstract

It is shown that if a partial latin square of order n with fewer than n entries has all its entries in no more than (n + 3) 2 rows, then it can be completed. This extends previous results of both Lindner and Wells, but lately Wells has improved this to (n + 5) 2 . We show that the number (n + 3) 2 is the best obtainable by the method of completing one row at a time without regard for completing future rows.