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For a poset $P=(X, \le_P)$, the linear discrepancy of $P$ is the minimum value of maximal differences of all incomparable elements for all possible labelings. In this paper, we find a lower bound and an upper bound of the linear discrepancy of a product of two posets. In order to give a lower bound, we use the known result, $\ld(\mathbf{m} \times \mathbf{n}) = \left\lceil \frac{mn}{2} \right\rceil -2$. Next, we use Dilworth's chain decomposition to obtain an upper bound of the linear discrepancy of a product of a poset and a chain. Finally, we give an example touching this upper bound.
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참고문헌 (9)
참고문헌 신청
M. Cheong / / Asymptotic results on linear discrepancy of the product of chains from its new bounds
M. Cheong / / The linear discrepancy of the product of three chains of the same size
M. Cheong / 2008 / The linear discrepancy of the product of three chains of size 2n / Far East J. Math. Sci. (FJMS) 30 (2) : 285 ~ 298
M. T. Keller / 2010 / Degree bounds for linear discrepancy of interval orders and disconnected posets / Discrete Math 310 (15-16) : 2198 ~ 2203
P. Fishburn / 2001 / Linear discrepancy and bandwidth / Order 18 (3) : 237 ~ 245
M. Cheong / / The linear discrepancy of the product of three chains of the same size
M. Cheong / 2008 / The linear discrepancy of the product of three chains of size 2n / Far East J. Math. Sci. (FJMS) 30 (2) : 285 ~ 298
M. T. Keller / 2010 / Degree bounds for linear discrepancy of interval orders and disconnected posets / Discrete Math 310 (15-16) : 2198 ~ 2203
P. Fishburn / 2001 / Linear discrepancy and bandwidth / Order 18 (3) : 237 ~ 245
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