On Kramer–Mesner matrix partitioning conjecture ⨿⨿

In 1977, Ganter and Teirlinck proved that any 2t×2t2t×2t matrix with 2t2t nonzero elements can be partitioned into four submatrices of order tt of which at most two contain nonzero elements. In 1978, Kramer and Mesner conjectured that any mt×ntmt×nt matrix with ktkt nonzero elements can be partitioned into mnmn submatrices of order tt of which at most kk contain nonzero elements. In 1995, Brualdi et al. showed that this conjecture is true if m=2m=2, k≤3k≤3 or k≥mn−2k≥mn−2. They also found a counterexample of this conjecture for m=4m=4, n=4n=4, k=6k=6 and t=2t=2. When t=2t=2, Rho showed that this conjecture is true if k≤5k≤5. When t=2t=2 and m=3m=3, we show that this conjecture is true if k=6k=6 or n≤3n≤3. As a result, we show that when t=2t=2, this conjecture is true if k=mn−3k=mn−3 also.