On the number of drawings of a combinatorial triangulation

TL;DR

The paper investigates how many geometric realizations a fixed combinatorial triangulation can admit on a planar point set. It develops two constructive lower-bound strategies—a simple $k$-nested regular triangulation on layered points yielding $\\Omega(1,259^n)$ drawings and a more powerful $(t,l)$-double-chain framework that achieves $\\Omega(1,31002235^n)$—and establishes an upper bound of $O(5,61^n)$ by relating drawings to polygonizations on the double chain. The work connects combinatorial triangulation enumeration with geometric realizations, leveraging entropy-based counting and polygonization results to bound the multiplicity of drawings. Together, these results advance understanding of how point configurations influence the realization count and inform potential bounds on $tr(n)$ via double-chain analyses.

Abstract

In 1962, Tutte provided a formula for the number of combinatorial triangulations, that is, maximal planar graphs with a fixed triangular face and $n$ additional vertices. In this note, we study how many ways a combinatorial triangulation can be drawn as geometric triangulation, that is, with straight-line segments, on a given point set in the plane. Our central contribution is that there exists a combinatorial triangulation with n vertices that can be drawn in at least $Ω(1,31^n)$ ways on a set of n points as different geometric triangulations. We also show an upper bound on the number of drawings of a combinatorial triangulation on the so-called double chain point set.

Figures (11)

• Figure 1: Combinatorial triangulations on 5 vertices. Only the first two are geometric triangulations.
• Figure 2: A combinatorial triangulation is drawn as two geometric triangulations on the same set of points.
• Figure 3: An example of a 3-nested regular triangulation.
• Figure 4: Rotation of an interior layer.
• Figure 5: Cases when 3 divides $n-1$ (left) and when 3 divides $n-2$ (right).

Theorems & Definitions (12)

• Definition 1
• Proposition 1
• Theorem 1
• Proposition 2
• Definition 2
• proof
• Definition 3
• Definition 4
• Remark 1
• Definition 5