ON KRAMER-MESNER MATRIX PARTITIONING CONJECTURE

Title & Authors
ON KRAMER-MESNER MATRIX PARTITIONING CONJECTURE
Rho, Yoo-Mi;

Abstract
In 1977, Ganter and Teirlinck proved that any $\small{2t\;\times\;2t}$ matrix with 2t nonzero elements can be partitioned into four sub-matrices of order t of which at most two contain nonzero elements. In 1978, Kramer and Mesner conjectured that any $\small{mt{\times}nt}$ 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 $\small{m = 2,\;k\;\leq\;3\;or\;k\geq\;mn-2}$. 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 $\small{k{\leq}5}$.邻�⨀Ȁ邻�⨀낻�⨀Ȁ낻�⨀킻�⨀Ѐ킻�⨀삻�⨀ࠀ삻�⨀𾶖⨀Ȁ𾶖⨀ᢼ�⨀Ȁᢼ�⨀ゼ�⨀Ȁゼ�⨀䢼�⨀⠀䢼�⨀낼�⨀Ȁ낼�⨀ᢽ�⨀Ȁᢽ�⨀肽�⨀Ȁ肽�⨀�⨀Ȁ�⨀룏�⨀Ā룏�⨀ꂐ�⨀Ԁꂐ�⨀傾�⨀܀傾�⨀墾�⨀Ԁ墾�⨀炾�⨀܀炾�⨀碾�⨀Ԁ碾�⨀棏�⨀Ȁ棏�⨀壏�⨀Ȁ壏�⨀䂐�⨀؀䂐�⨀䃏�⨀Ȁ䃏�⨀惏�⨀Ȁ惏�⨀墐�⨀Ȁ墐�⨀ꃏ�⨀Āꃏ�⨀⢐�⨀Ȁ⢐�⨀႐�⨀Ȁ႐�⨀�⨀Ȁ�⨀�⨀Ȁ
Keywords
partition of matrices;bipartite graphs;adjacency matrices;matrix-crossings;P$\small{^3}$-claws;trees;unicyclic graphs;
Language
English
Cited by
1.
On Kramer–Mesner matrix partitioning conjecture ⨿, Discrete Mathematics, 2010, 310, 12, 1793
References
1.
R. A. Brualdi et al., On a matrix partition conjecture, J. Combin. Theory Ser. A 69 (1995), no. 2, 333-346

2.
P. Erdos, A. Ginzburg, and A. Ziv, Bull. Res. Council Israel, 10F (1961), 41-43

3.
B. Ganter and L. Terlinck, A combinatorial lemma, Math. Z. 154 (1977), 153-156

4.
J. E. Olson, A combinatorial problem on finite abelian groups 1 and 2, J. Number Theory 1 (1969), 8-10 and 195-199

5.
I. Reiman, Uber ein Problem von K. Zarankiewicz, Acta. Math. Acad. Sci. Hungar. 9 (1958), 269-279