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.

We prove that every n-vertex 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.

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 0001-8708#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. Humboldt-Stiftung while he was at the Institut fu# r diskrete Mathematik, Bonn; and by the U. S. Office of Naval Research under the contract N-0014-91-J-1385. 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 2-manifold is topologically equivalent either to a sphere with g#0 handles, S g (orientable surface with genus g), or to a sphere

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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 polynomial-time 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.

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 h-matrix of the arrangement (a generalization, introduced by Mulmuley, of the h-vector 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 h-matrix also imply bounds for the number of i-dimensional 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.

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., Vrt'o, I., On 3-layer 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 two-layered 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), 369-376. [217] Sarazin, M.L., The crossing number of the genera

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