Latin squares and their defining sets

A Latin square L(n, k) is a square of order n with its entries colored with k colors so that all the entries in a row or column have different colors. Let d(L(n, k)) be the minimal number of colored entries of an n × n square such that there is a unique way of coloring of the yet uncolored entries in order to obtain a Latin square L(n, k). In this paper we discuss the properties of d(L(n, k)) for k = 2n − 1 and k = 2n − 2. We give an alternate proof of the identity d(L(n, 2n− 1)) = n − n, which holds for even n, and we establish the new result d(L(n, 2n− 2)) ≥ n − ⌊ 8n 5 ⌋ and show that this bound is tight for n divisible by 10.