On kramer-mesner matrix partitioning conjecture

Abstract

In 1977, Ganter and Teirlinck proved that any 2t × 2t matrix with 2t nonzero elements can be partitioned into four submatrices of order t of which at most two contain nonzero elements. In 1978, Kramer and Mesner conjectured that any mt × nt matrix with kt nonzero elements can be partitioned into mn submatrices of order t of which at most k contain nonzero elements. In 1995, Brualdi et al. showed that this conjecture is true if m = 2, k ≤ 3 or k ≥ mn - 2. They also found a counterexample of this conjecture when m = 4, n = 4, k = 6 and t = 2. When t = 2, we show that this conjecture is true if k ≤ 5.