Crossings in Grid Drawings

We prove crossing number inequalities for geometric graphs whose vertex sets are taken from a d-dimensional grid of volume N and give applications of these inequalities to counting the number of non-crossing geometric graphs that can be drawn on such grids. In particular, we show that any geometric graph with m >= 8N edges and with vertices on a 3D integer grid of volume N, has \Omega((m^{2/n)\log(m/n))} crossings. In d-dimensions, with d >= 4, this bound becomes \Omega(m^2/n). We provide matching upper bounds for all d. Finally, for d >= 4 the upper bound implies that the maximum number of crossing-free geometric graphs with vertices on some d-dimensional grid of volume N is n^\Theta(n). In 3 dimensions it remains open to improve the trivial bounds, namely, the 2^\Omega(n) lower bound and the n^O(n) upper bound.