On the separation power and the completion of partial latin squares

Abstract

Let n and m be natural numbers, n ⩾ m. The separation power of order n and degree m is the largest integer k = k(n, m) such that for every (0, 1)-matrix A of order n with constant linesums equal to m and any set of k 1's in A there exist (disjoint) permutation matrices P1,…, Pm such that A = P1 + … + Pm and each of the k 1's lies in a different Pi. Almost immediately we have 1 ⩽ k(n, m) ⩽ m − 1, yet in all cases where the value of k(n, m) is actually known it equals m − 1 (except under the somewhat trivial circumstances of k(n, m) = 1). This leads to a conjecture about the separation power, namely that k(n, m) = m − 1 if m ⩾ [n2] + 1. We obtain the bound k(n, m) ⩾ m − [n2] + 2, so that this conjecture holds for n ⩽ 7. We then move on to latin squares, describing several equivalent formulations of the concept. After establishing a sufficient condition for the completion of a partial latin square in terms of the separation power, we can show that the Evans conjecture follows from this conjecture about the separation power. Finally the lower bound on k(n, m) allows us to show, after some calculations, that the Evans conjecture is true for orders n ⩽ 11.