k-Sets, Convex Quadrilaterals, and the Rectilinear Crossing Number of Kn
نویسندگان
چکیده
منابع مشابه
k-Sets, Convex Quadrilaterals, and the Rectilinear Crossing Number of Kn
We use circular sequences to give an improved lower bound on the minimum number of (≤ k)– sets in a set of points in general position. We then use this to show that if S is a set of n points in general position, then the number (S) of convex quadrilaterals determined by the points in S is at least 0.37533 ` n 4 ́ + O(n). This in turn implies that the rectilinear crossing number cr(Kn) of the com...
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We use circular sequences to give an improved lower bound on the minimum number of (≤ k)-sets in a set of points in general position. We then use this to show that if S is a set of n points in general position, then the number (S) of convex quadrilaterals determined by the points in S is at least 0.37553 ( n 4 ) +O(n). This in turn implies that the rectilinear crossing number cr(Kn) of the comp...
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We provide a new lower bound on the number of (≤ k)-edges of a set of n points in the plane in general position. We show that for 0 ≤ k ≤ bn−2 2 c the number of (≤ k)-edges is at least Ek(S) ≥ 3 ( k + 2 2 ) + k ∑ j=b3 c (3j − n + 3), which, for b3 c ≤ k ≤ 0.4864n, improves the previous best lower bound in [11]. As a main consequence, we obtain a new lower bound on the rectilinear crossing numbe...
متن کاملRecent developments on the number of ( ≤ k ) - sets , halving lines , and the rectilinear crossing number of Kn . Bernardo
We present the latest developments on the number of (≤ k)-sets and halving lines for (generalized) configurations of points; as well as the rectilinear and pseudolinear crossing numbers of Kn. In particular, we define perfect generalized configurations on n points as those whose number of (≤ k)-sets is exactly 3¡k+1 2 ¢ for all k ≤ n/3. We conjecture that for each n there is a perfect configura...
متن کاملConvex Quadrilaterals and k-Sets
We prove that the minimum number of convex quadrilaterals determined by n points in general position in the plane – or in other words, the rectilinear crossing number of the complete graph Kn – is at least ( 38 + 10 −5) ( n 4 ) +O(n). Our main tool is a lower bound on the number of (≤ k)-sets of the point set: we show that for every k ≤ n/2, there are at least 3 ( k+1 2 ) subsets of size at mos...
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ژورنال
عنوان ژورنال: Discrete & Computational Geometry
سال: 2006
ISSN: 0179-5376,1432-0444
DOI: 10.1007/s00454-005-1227-6