On Completing Latin Squares

Abstract

This paper characterizes a particular scheme of partially filled Latin squares and when they can be completed to full Latin squares. In particular, given an $n\times n$ array with the first $s$ rows and the first $d$ cells of row $s+1$ filled with $n$ distinct symbols in such a way that no symbol occurs more than once in any row or column, necessary and sufficient conditions are found for when this array can be completed to a full Latin square.