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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.
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 8 (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
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 7 (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.
On a Geometric Generalization of the Upper Bound Theorem
"... We prove an upper bound, tight up to a factor of 2, for the number of vertices of level at most ℓ in an arrangement of n halfspaces in R d, for arbitrary n and d (in particular, the dimension d is not considered constant). This partially settles a conjecture of Eckhoff, Linhart, and Welzl. Up to the ..."
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Cited by 3 (1 self)
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We prove an upper bound, tight up to a factor of 2, for the number of vertices of level at most ℓ in an arrangement of n halfspaces in R d, for arbitrary n and d (in particular, the dimension d is not considered constant). This partially settles a conjecture of Eckhoff, Linhart, and Welzl. Up to the factor of 2, the result generalizes McMullen’s Upper Bound Theorem for convex polytopes (the case ℓ = 0) and extends a theorem of Linhart for the case d ≤ 4. Moreover, the bound sharpens asymptotic estimates obtained by Clarkson and Shor. The proof is based on the hmatrix of the arrangement (a generalization, introduced by Mulmuley, of the hvector of a convex polytope). We show that bounding appropriate sums of entries of this matrix reduces to a lemma about quadrupels of sets with certain intersection properties, and we prove this lemma, up to a factor of 2, using tools from multilinear algebra. This extends an approach of Alon and Kalai, who used linear algebra methods for an alternative proof of the classical Upper Bound Theorem. The bounds for the entries of the hmatrix also imply bounds for the number of idimensional faces, i> 0, at level at most ℓ. Furthermore, we discuss a connection with crossing numbers of graphs that was one of the main motivations for investigating exact bounds that are valid for arbitrary dimensions. 1.
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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Cited by 2 (1 self)
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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
Crossing Numbers of Graphs: A Bibliography
"... ., Vrt'o, I., On 3layer crossings and pseudo arrangements, in: Proc. 7th Symposium on Graph Drawing, Lecture Notes in Computer Science, Springer Verlag, Berlin, 1999. [252] Mutzel, P., Ziegler, T., The constrained crossing minimization problem, in: Proc. 7th Symposium on Graph Drawing, Lecture Not ..."
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., Vrt'o, I., On 3layer crossings and pseudo arrangements, in: Proc. 7th Symposium on Graph Drawing, Lecture Notes in Computer Science, Springer Verlag, Berlin, 1999. [252] Mutzel, P., Ziegler, T., The constrained crossing minimization problem, in: Proc. 7th Symposium on Graph Drawing, Lecture Notes in Computer Science, Springer Verlag, Berlin, 1999. [253] Yamaguchi, A., Sugimoto, A., An approximation algorithm for the twolayered graph drawing problem, in: Proc. Combinatorics and Computing, COCOON99, Lecture Notes in Computer Science 1627, Springer Verlag, Berlin, 1999. 10 [215] Kimm, Parallel total weight crossing number on a linear array with a reconfigurable pipeline bus system, 29th Southeastern Symposium on System Theory'97. [216] Lov'asz, Pach, J., Szegedy, M., On Conway's thrackle conjecture, Discrete and computational Geometry 18 (1997), 369376. [217] Sarazin, M.L., The crossing number of the genera