Abstract

The full n-Latin square is the $$n\times n$$n×n array with symbols $$1,2,\dots ,n$$1,2,?,n in each cell. In a way that is analogous to critical sets of full designs, a critical set of the full n-Latin square can be used to find a defining set for any Latin square of order n. In this paper we study the size of the smallest critical set for a full n-Latin square, showing this to be somewhere between $$(n^3-2n^2+2n)/2$$(n3-2n2+2n)/2 and $$(n-1)^3+1$$(n-1)3+1. In the case that each cell is either full or empty, we show the size of a critical set in the full n-Latin square is always equal to $$n^3-2n^2-n$$n3-2n2-n.

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