On the completion of partial latin squares

Abstract

In this paper a certain condition on partial latin squares is shown to be sufficient to guarantee that the partial square can be completed, namely, that it have fewer than n entries, and that at most [ (n + 1) 2 ] of these lie off some line, where n is the order of the square. This is applied to establish that the Evans conjecture is true for n ⩽ 8; i.e., that given a partial latin square of order n with fewer than n entries, n ⩽ 8, the square can be completed. Finally, the results are viewed in a conjugate way to establish different conditions sufficient for the completion of a partial latin square.