Counting Triangulations of Planar Point Sets

TL;DR

This paper establishes a tighter upper bound of 30^n on the maximum number of triangulations for any set of n planar points, improving previous bounds and aiding in bounding other planar graph types.

Contribution

It introduces a refined charging scheme to significantly improve the upper bound on triangulations from 43^n to 30^n, and derives new bounds for various planar graphs.

Findings

Maximum triangulations bounded by 30^n

New bounds for planar graphs, spanning cycles, and spanning trees

Improved understanding of planar straight-line graph counts

Abstract

We study the maximal number of triangulations that a planar set of $n$ points can have, and show that it is at most $30^n$. This new bound is achieved by a careful optimization of the charging scheme of Sharir and Welzl (2006), which has led to the previous best upper bound of $43^n$ for the problem. Moreover, this new bound is useful for bounding the number of other types of planar (i.e., crossing-free) straight-line graphs on a given point set. Specifically, we derive new upper bounds for the number of planar graphs ($o(239.4^n)$), spanning cycles ($O(70.21^n)$), and spanning trees ($160^n$).