Completing Partial Latin Squares with Blocks of Non-empty Cells

Abstract

In this paper we develop two methods for completing partial latin squares and prove the following. Let $A$ be a partial latin square of order $nr$ in which all non-empty cells occur in at most $n-1$ $r\times r$ squares. If $t_1,\ldots , t_m$ are positive integers for which $n\geqslant t_1^2+t_2^2+\cdots +t_m^2+1$ and if $A$ is the union of $m$ subsquares each with order $rt_i$, then $A$ can be completed. We additionally show that if $n\geqslant r+1$ and $A$ is the union of $n$ identical $r\times r$ squares with disjoint rows and columns, then $A$ can be completed. For smaller values of $n$ we show that a completion does not always exist.