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Crossing Numbers: Bounds and Applications
 I. B'AR'ANY AND K. BOROCZKY, BOLYAI SOCIETY MATHEMATICAL STUDIES 6
, 1997
"... We give a survey of techniques for deriving lower bounds and algorithms for constructing upper bounds for several variations of the crossing number problem. Our aim is to emphasize the more general results or those results which have an algorithmic flavor, including the recent results of the autho ..."
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We give a survey of techniques for deriving lower bounds and algorithms for constructing upper bounds for several variations of the crossing number problem. Our aim is to emphasize the more general results or those results which have an algorithmic flavor, including the recent results of the authors. We also show applications of crossing numbers to other areas of discrete mathematics, like discrete geometry.
Edge Separators For Graphs Of Bounded Genus With Applications
, 1993
"... We prove that every nvertex graph of genus g and maximal degree k has an edge separator of size O( gkn). The upper bound is best possible to within a constant factor. This extends known results on planar graphs and similar results about vertex separators. We apply the edge separator to the is ..."
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Cited by 8 (1 self)
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We prove that every nvertex graph of genus g and maximal degree k has an edge separator of size O( gkn). The upper bound is best possible to within a constant factor. This extends known results on planar graphs and similar results about vertex separators. We apply the edge separator to the isoperimetric number problem, graph embeddings and lower bounds for crossing numbers.
The Crossing Number of a Graph on a Compact 2Manifold
, 1996
"... INTRODUCTION We assume that the reader is familiar with the basic concepts of graph theory as in [CL86] and the basic concepts of topological graph theory as in [WB78]. By the famous theorem of Brahana [Br23], any compact article no. 0069 105 00018708#96 #18.00 Copyright # 1996 by Academic Press ..."
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Cited by 7 (4 self)
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INTRODUCTION We assume that the reader is familiar with the basic concepts of graph theory as in [CL86] and the basic concepts of topological graph theory as in [WB78]. By the famous theorem of Brahana [Br23], any compact article no. 0069 105 00018708#96 #18.00 Copyright # 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. * Research of the third author was supported by the A. v. HumboldtStiftung while he was at the Institut fu# r diskrete Mathematik, Bonn; and by the U. S. Office of Naval Research under the contract N001491J1385. File: 607J 155202 By:CV . Date:21:11:96 Time:08:36 LOP8M. V8.0. Page 01:01 Codes: 3060 Signs: 2268 Length: 45 pic 0 pts, 190 mm 2manifold is topologically equivalent either to a sphere with g#0 handles, S g (orientable surface with genus g), or to a sphere
Improved lower bounds for the 2page crossing numbers of Km,n and Kn via semidefinite programming
, 2011
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Drawings of Graphs on Surfaces with Few Crossings
, 1996
"... . We give drawings of a complete graph K n with O(n 4 log 2 g/g) many crossings on an orientable or nonorientable surface of genus g # 2. We use these drawings of K n and give a polynomialtime algorithm for drawing any graph with n vertices and m edges with O(m 2 log 2 g/g) many crossings ..."
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Cited by 3 (0 self)
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. We give drawings of a complete graph K n with O(n 4 log 2 g/g) many crossings on an orientable or nonorientable surface of genus g # 2. We use these drawings of K n and give a polynomialtime algorithm for drawing any graph with n vertices and m edges with O(m 2 log 2 g/g) many crossings on an orientable or nonorientable surface of genus g # 2. Moreover, we derive lower bounds on the crossing number of any graph on a surface of genus g # 0. The number of crossings in the drawings produced by our algorithm are within a multiplicative factor of O(log 2 g) from the lower bound (and hence from the optimal) for any graph with m # 8n and n 2 /m # g # m/64. Key Words. Crossing number, Orientable and nonorientable surface, Topological graph theory. 1.
The 2page crossing number of Kn
, 2011
"... Around 1958, Hill conjectured that the crossing number cr(Kn) of the complete graph Kn is Z (n): = 1 n n−1 n−2 n−3 4 2 2 2 2 and provided drawings of Kn with exactly Z(n) crossings. Towards the end of the century, substantially different drawings of Kn with Z(n) crossings were found. These drawings ..."
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Around 1958, Hill conjectured that the crossing number cr(Kn) of the complete graph Kn is Z (n): = 1 n n−1 n−2 n−3 4 2 2 2 2 and provided drawings of Kn with exactly Z(n) crossings. Towards the end of the century, substantially different drawings of Kn with Z(n) crossings were found. These drawings are 2page book drawings, that is, drawings where all the vertices are on a line ℓ (the spine) and each edge is fully contained in one of the two halfplanes (pages) defined by ℓ. The2page crossing number of Kn, denoted by ν2(Kn), is the minimum number of crossings determined by a 2page book drawing of Kn. It was generally conjectured that cr(Kn) =Z(n) and since cr(Kn) ≤ ν2(Kn) ≤ Z(n), the conjecture ν2(Kn) =Z(n) appeared as a milestone in the way to find the correct values of cr(Kn). In this paper we develop a novel and innovative technique to investigate crossings in drawings of Kn, and use it to prove that ν2(Kn) =Z(n). To this end, we extend the inherent geometric definition of kedges for finite sets of points in the plane to topological drawings of Kn. We also introduce the concept of ≤≤kedges as a useful generalization of ≤kedges. Finally, we extend a powerful theorem that expresses the number of crossings in a rectilinear drawing of Kn in terms of its number of kedges to the topological setting. 1 1
On a geometric generalization of the upper bound theorem
 In FOCS: IEEE Symposium on Foundations of Computer Science (FOCS
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