Number of Crossing-Free Geometric Graphs vs. Triangulations

We show that there is a constant α > 0 such that, for any set P of n⩾ 5 points in general position in the plane, a crossing-free geometric graph on P that is chosen uniformly at random contains, in expectation, at least ( 1 2 + α ) M edges, where M denotes the number of edges in any triangulation of P. From this we derive (to our knowledge) the first non-trivial upper bound of the form c n ⋅ tr ( P ) on the number of crossing-free geometric graphs on P; that is, at most a fixed exponential in n times the number of triangulations of P. (The trivial upper bound of 2 M ⋅ tr ( P ) , or c = 2 M / n , follows by taking subsets of edges of each triangulation.) If the convex hull of P is triangular, then M = 3 n − 6 , and we obtain c < 7.98 . bounds for the number of crossing-free geometric graphs on planar point sets have so far applied the trivial 8 n factor to the bound for triangulations; we slightly decrease this bound to O ( 343.11 n ) .