New Lower Bounds for the Number of (≤ k)-Edges and the Rectilinear Crossing Number of Kn
نویسندگان
چکیده
منابع مشابه
New Lower Bounds for the Number of (<=k)-Edges and the Rectilinear Crossing Number of Kn
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...
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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 ≤ ⌊ 2 ⌋ the number of (≤ k)-edges is at least Ek(S) ≥ 3 (
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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 ≤ ⌊ 2 ⌋ the number of (≤ k)-edges is at least Ek(S) ≥ 3 (
متن کاملToward the rectilinear crossing number of Kn: new drawings, upper bounds, and asymptotics
Scheinerman and Wilf [SW94] assert that “an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph Kn.” A rectilinear drawing of Kn is an arrangement of n vertices in the plane, every pair of which is connected by an edge that is a line segment. We assume that no three vertices are collinear, and that no three edges intersec...
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ژورنال
عنوان ژورنال: Discrete & Computational Geometry
سال: 2007
ISSN: 0179-5376,1432-0444
DOI: 10.1007/s00454-007-1325-8