Crossing Numbers and Combinatorial Characterization of Monotone Drawings of $$K_n$$ K n
نویسندگان
چکیده
In 1958, Hill conjectured that the minimum number of crossings in a drawing of Kn is exactly Z(n) = 1 4 ⌊n 2 ⌋ ⌊
منابع مشابه
The crossing numbers of $K_{n,n}-nK_2$, $K_{n}\times P_2$, $K_{n}\times P_3$ and $K_n\times C_4$
The crossing number of a graph G is the minimum number of pairwise intersections of edges among all drawings of G. In this paper, we study the crossing number of Kn,n − nK2, Kn × P2, Kn × P3 and Kn × C4.
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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...
متن کاملMore on the crossing number of Kn: Monotone drawings
The Harary-Hill conjecture states that the minimum number of crossings in a drawing of the complete graph Kn is Z(n) := 1 4 ⌊ n 2 ⌋ ⌊ n−1 2 ⌋ ⌊ n−2 2 ⌋ ⌊ n−3 2 ⌋ . This conjecture was recently proved for 2-page book drawings of Kn. As an extension of this technique, we prove the conjecture for monotone drawings of Kn, that is, drawings where all vertices have different x-coordinates and the edg...
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A geometric graph is a graph G = (V;E) drawn in the plane so that the vertex set V consists of points in general position and the edge set E consists of straight line segments between points of V . It is known that, for any xed k, any geometric graph G on n vertices with no k pairwise crossing edges contains at most O(n log n) edges. In this paper we give a new, simpler proof of this bound, and...
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ورودعنوان ژورنال:
- Discrete & Computational Geometry
دوره 53 شماره
صفحات -
تاریخ انتشار 2015