Shellable Drawings and the Cylindrical Crossing Number of $$K_{n}$$ K n
نویسندگان
چکیده
منابع مشابه
Shellable drawings and the cylindrical crossing number of $K_n$
The Harary-Hill Conjecture states that the number of crossings in any drawing of the complete graph Kn in the plane is at least Z(n) := 1 4 ⌊ n 2 ⌋ ⌊ n−1 2 ⌋ ⌊ n−2 2 ⌋ ⌊ n−3 2 ⌋ . In this paper, we settle the Harary-Hill conjecture for shellable drawings. We say that a drawing D of Kn is s-shellable if there exist a subset S = {v1, v2, . . . , vs} of the vertices and a region R of D with the fo...
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The Harary-Hill conjecture states that for every n > 0 the complete graph on n vertices Kn, the minimum number of crossings over all its possible drawings equals H(n) := 1 4 ⌊n 2 ⌋⌊n− 1 2 ⌋⌊n− 2 2 ⌋⌊n− 3 2 ⌋ . So far, the lower bound of the conjecture could only be verified for arbitrary drawings of Kn with n ≤ 12. In recent years, progress has been made in verifying the conjecture for certain ...
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In 1958, Hill conjectured that the minimum number of crossings in a drawing of Kn is exactly Z(n) = 1 4 ⌊n 2 ⌋ ⌊
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The best lower bound known on the crossing number of the complete bipartite graph is : cr(Km,n) ≥ (1/5)(m)(m − 1)bn/2cb(n − 1)/2c In this paper we prove that: cr(Km,n) ≥ (1/5)m(m − 1)bn/2cb(n − 1)/2c + 9.9 × 10−6m2n2 for sufficiently large m and n.
متن کاملThe circular k-partite crossing number of Km, n
We define a new kind of crossing number which generalizes both the bipartite crossing number and the outerplanar crossing number. We calculate exact values of this crossing number for many complete bipartite graphs and also give a lower bound. 1 Preliminaries The bipartite crossing number of a bipartite graph G was defined by Watkins in [W] to be the minimum number of crossings over all biparti...
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ژورنال
عنوان ژورنال: Discrete & Computational Geometry
سال: 2014
ISSN: 0179-5376,1432-0444
DOI: 10.1007/s00454-014-9635-0