On the crossing numbers of loop networks and generalized Petersen graphs Original Research Article

Bhatt and Leighton proved that the crossing number of a network (graph) is closely related to the minimum layout area required for the implementation of a VLSI circuit for that network. With this important application in mind, it makes most sense to analyze the crossing numbers of graphs with good interconnection properties, such as the circulant graphs image. In this work we find tight bounds for the crossing numbers of the (double fixed step) circulant graphs image. Specifically, we show that for values of n sufficiently large compared to k, the crossing number of image is bounded by above and by below by linear functions of n, both of which have coefficients that approach 1 as k goes to infinity. As an additional application of these bounds, we show that the crossing numbers of the generalized Petersen graphs image are bounded by below and by above by linear functions of n, whose coefficients approach image and 2, respectively, as k goes to infinity.