. We prove that the minimal VLSI layout of the arrangement graph A(n; k) occupies \Theta(n!=(n \Gamma k \Gamma 1)!) 2 area. As a special case we obtain an optimal layout for the star graph S n with the area \Theta(n!) 2 : This answers an open problem posed by Akers, Harel and Krishnamurthy [1]. The method is also applied to the pancake graph. The results provide optimal upper and lower bounds for crossing numbers of the above graphs. Key Words: area, arrangement graph, congestion, crossing number, embedding, layout, pancake graph, star graph, VLSI 1

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.

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

. Let Q n denote the n--dimensional cube. In this paper, we exhibit drawings for n = 6, 7 and 8. In these cases the drawings confirm Eggleton and Guy's conjectured upper bound for the crossing number of the n--cube. 1 Introduction A simple drawing D(G) of a graph G(V; E) is a drawing of G in the plane, in which no edge crosses itself, adjacent edges do not cross, crossing edges do so only once, edges do not cross nodes, and no more than two edges cross at a common node. In what follows, all drawings are assumed to be simple. A simple drawing is optimum if it has the minimum number crossings; this number is called the crossing number of G and is denoted by (G). The algorithmic problem of computing the crossing number of a graph has been shown to be NP-complete [GaJo 83]. Let Q n denote the n--cube. The nodes of Q n are all n--tuples of 0's and 1's of which there are N = 2 n . Nodes x = (x 1 ; x 2 ; : : : ; x n ) and y = (y 1 ; y 2 ; : : : ; y n ) are adjacent if and only if x i 6= ...

The splitting number of a graph G consists in the smallest positive integer k 0, such that a planar graph can be obtained from G by k splitting operations, such operation replaces v by two nonadjacent vertices v1 and v2 , and attaches the neighbors of v either to v1 or to v2 . One of the most useful graphs in computer science is the n--cube. Dean and Richter devoted an article to proving that the minimum number of crossings in an optimum drawing of the 4--cube is 8, but no results about splitting number of nonplanar n--cubes are known. In this note we give a proof that the splitting number of the 4--cube is 4. In addition, we give the lower bound 2 n\Gamma2 for the splitting number of the n--cube. In particular, because it is known that the splitting number of the n--cube is O(2 n ), our result implies that the splitting number of the n-cube is \Theta(2 n ).

. 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.

The minor crossing number of a graph G, mcr(G), is defined as the minimum crossing number of all graphs that contain G as a minor. We present some basic properties of this new minor-monotone graph invariant. We give

Interconnection networks play a vital role in parallel computing architectures. We investigate topological properties of some networks proposed for parallel computation, based on their underlying graph models. The vertices of the graph correspond to processors and the edges represent communication links between processors. Parameters such as crossing number and thickness strongly affect the area required to lay out the corresponding circuit on a VLSI chip. In particular, we give upper bounds for the skewness, crossing number, and thickness of several networks including the mesh of trees, reduced mesh of trees, 2-dimensional torus, butterfly, wrapped butterfly, and Benes graph.

We study the integral uniform (multicommodity) flow problem in a graph G and construct a fractional solution whose properties are invariant under the action of a group of automorphisms # < Aut(G). The fractional solution is shown to be close to an integral solution (depending on properties of #), and in particular becomes an integral solution for a class of graphs containing Cayley graphs. As an application we estimate asymptotically (up to additive error terms) the edge congestion of an optimal integral uniform flow (edge forwarding index) in the cube connected cycles and the butterfly. # The research of the first author was supported by NSF grant CCR-9528228. Research of the second author was supported in part by the Hungarian NSF contracts T 016 358 and T 019 367, and by the NSF contract DMS 970 1211. 1 1 Introduction The uniform concurrent multicommodity flow (uniform flow) problem [14, 17] is the problem of supplying one unit of (fractional) flow between all ordered pairs of v...

In this paper we obtain bounds on the area and wire length required by VLSI layouts of homogeneous product networks with any number of dimensions. The lower bounds are obtained by computing lower bounds on the bisection width and the crossing number. The upper bounds are derived by using traditional frameworks like separators and bifurcators, as well as a new method based on combining collinear layouts. This last method has led to the best area and wire lengths for most of the homogeneous product networks we considered. 1: Introduction The cross product is a well known operation defined on graphs. When applied to interconnection networks, the cross product operation combines a set of "factor" networks into a product network. Several well known networks are instances of product networks, including the grid, the torus, and the hypercube. A product network is said to be homogeneous if all its factor networks are isomorphic. Otherwise the product network is heterogeneous. All the a...