Bounds on the Maximum Multiplicity of Some Common Geometric Graphs

We obtain new lower and upper bounds for the maximum multiplicity of some weighted and, respectively, nonweighted common geometric graphs drawn on $n$ points in the plane in general position (with no three points collinear): perfect matchings, spanning trees, spanning cycles (tours), and triangulations. (i) We present a new lower bound construction for the maximum number of triangulations a set of $n$ points in general position can have. In particular, we show that a generalized double chain formed by two almost convex chains admits $\Omega (8.65^n)$ different triangulations. This improves the bound $\Omega (8.48^n)$ achieved by the previous best construction, the double zig-zag chain studied by Aichholzer et al. (ii) We obtain a new lower bound of $\Omega(12.00^n)$ for the number of noncrossing spanning trees of the double chain composed of two convex chains. The previous bound, $\Omega(10.42^n)$, stood unchanged for more than 10 years. (iii) Using a recent upper bound of $30^n$ for the number of triangu...