We prove tight 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 crossing-free 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 Ω((m2/N) log(m/N)) crossings. In d-dimensions, with d ≥ 4, this bound becomes Ω(m2/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 NΘ(N). In 3 dimensions it remains open to improve the trivial bounds, namely, the 2Ω(N) lower bound and the NO(N) upper bound.

Additional Metadata
Keywords Graph drawing, Grid drawing
Journal Electronic Journal of Combinatorics
Dujmović, V, Morin, P, & Sheffer, A. (Adam). (2014). Crossings in grid drawings. Electronic Journal of Combinatorics, 21(1).