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ON KRAMER-MESNER MATRIX PARTITIONING CONJECTURE
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 Title & Authors
ON KRAMER-MESNER MATRIX PARTITIONING CONJECTURE
Rho, Yoo-Mi;
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 Abstract
In 1977, Ganter and Teirlinck proved that any 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 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 . 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 .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-claws;trees;unicyclic graphs;
 Language
English
 Cited by
1.
On Kramer–Mesner matrix partitioning conjecture ⨿, Discrete Mathematics, 2010, 310, 12, 1793  crossref(new windwow)
 References
1.
R. A. Brualdi et al., On a matrix partition conjecture, J. Combin. Theory Ser. A 69 (1995), no. 2, 333-346 crossref(new window)

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 crossref(new window)

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

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