Critical sets in back circulant Latin rectangles

A latin rectangle is an m × n array, m ≤ n, from the numbers 1, 2, . . . , n such that each of these numbers occur in each row and in each column at most once. A critical set in an m×n array is a set S of given entries, such that there exists a unique extension of S to a latin rectangle of size m×n. If we index the rows and columns of an m×n array, m ≤ n, by the sets M = {1, 2, . . . ,m} and N = {1, 2, . . . , n}, respectively, then the array with integer i+j−1 (mod n) in the position (i, j) is said to be a back circulant latin rectangle. We show that the size of smallest critical set in a back circulant latin rectangle of size m× n, with 4m ≤ 3n is equal to m(n−m) + b(m− 1)2/4c.