Critical sets of 2-balanced Latin rectangles

Abstract

An (m, n, 2)-balanced Latin rectangle is an (Formula presented.) 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, (Formula presented.), the minimum size for such a structure is shown to be (Formula presented.). Such critical sets in turn determine defining sets for (0, 1)-matrices.