Journal of the
Korean Mathematical Society JKMS

In $1977$, Ganter and Teirlinck proved that any $2t \times 2t$ 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 $mt \times nt$ 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 $m = 2$, $k \leq 3$ or $k \geq mn-2$. 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 $k \leq 5$.

Keywords: partition of matrices, bipartite graphs, adjacency matrices, matrix-crossings, $P_3$-claws, trees, unicyclic graphs

MSC numbers: Primary 05C50; Secondary 05C35, 15A23