Critical Sets of 2-Balanced Latin Rectangles

Abstract

An (m, n, 2)-balanced Latin rectangle is an m × n array on symbols 0 and 1 such that each symbol occurs n times in each row and m times in each column, with each cell containing either two 0's, two 1's or both 0 and 1. We completely determine the structure of all critical sets of the full (m, n, 2)-balanced Latin rectangle (which contains 0 and 1 in each cell). If m, n ≥ 2, the minimum size for such a structure is shown to be (m − 1)(n − 1) + 1. Such critical sets in turn determine defining sets for (0, 1)-matrices.

Lower Bounds on the Sizes of Defining Sets in Full n-Latin Squares and Full Designs

“…A defining set D of an (m, n, t)-balanced Latin rectangle L is a partial (n, m, t)-balanced Latin rectangle with unique completion to L. In the example below, the partial (2, 3, 3)-balanced Latin rectangle (on the left) is not a defining set for F 2,3,3 (in the centre) as it also completes to another (2,3,3)-balanced Latin rectangle (on the right).…”

“…The bound presented is unlikely to be exact; in [4] defining sets of size (n − 1) 3 + 1 of the full n-Latin square are constructed for n 2; this remains the smallest known construction. The structure of minimal defining sets of F m,n,2 is completely determined in [3].…”