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ON KRAMER-MESNER MATRIX PARTITIONING CONJECTURE

  • Rho, Yoo-Mi (Department of Mathematics Incheon University)
  • 발행 : 2005.07.01

초록

In 1977, Ganter and Teirlinck proved that any $2t\;\times\;2t$ 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 $mt{\times}nt$ 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 $k{\leq}5$.

키워드

참고문헌

  1. R. A. Brualdi et al., On a matrix partition conjecture, J. Combin. Theory Ser. A 69 (1995), no. 2, 333-346 https://doi.org/10.1016/0097-3165(95)90056-X
  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 https://doi.org/10.1007/BF01241828
  4. J. E. Olson, A combinatorial problem on finite abelian groups 1 and 2, J. Number Theory 1 (1969), 8-10 and 195-199 https://doi.org/10.1016/0022-314X(69)90021-3
  5. I. Reiman, Uber ein Problem von K. Zarankiewicz, Acta. Math. Acad. Sci. Hungar. 9 (1958), 269-279 https://doi.org/10.1007/BF02020254

피인용 문헌

  1. On Kramer–Mesner matrix partitioning conjecture ⨿ vol.310, pp.12, 2010, https://doi.org/10.1016/j.disc.2009.12.003