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 $$A$$A be a partial latin square of order $$nr$$nr in which all non-empty cells occur in at most $$n-1$$n-1$$r\times r$$r×r squares. If $$t_1,\ldots , t_m$$t1,?,tm are positive integers for which $$n\geqslant t_1^2+t_2^2+\cdots +t_m^2+1$$n?t12+t22+?+tm2+1 and if $$A$$A is the union of $$m$$m subsquares each with order $$rt_i$$rti, then $$A$$A can be completed. We additionally show that if $$n\geqslant r+1$$n?r+1 and $$A$$A is the union of $$n$$n identical $$r\times r$$r×r squares with disjoint rows and columns, then $$A$$A can be completed. For smaller values of $$n$$n we show that a completion does not always exist.