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ON KRAMER-MESNER MATRIX PARTITIONING CONJECTURE

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

Abstract

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 $k{\leq}5$.

Keywords

References

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Cited by

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