[Paper Review] Counting Plane Graphs: Perfect Matchings, Spanning Cycles, and Kasteleyn's Technique

Micha Sharir, Adam Sheffer+1 more|arXiv (Cornell University)|Sep 26, 2011

Advanced Combinatorial Mathematics17 references4 citations

TL;DR

This paper improves the upper bound on the number of crossing-free straight-edge Hamiltonian cycles (simple polygonizations) that can be embedded over a set of N points in the plane, using Kasteleyn’s linear algebra technique and edge-flipping analysis. The key result is an improved bound of O(54.543^N), reducing the previous best bound of O(68.664^N).

ABSTRACT

We derive improved upper bounds on the number of crossing-free straight-edge spanning cycles (also known as Hamiltonian tours and simple polygonizations) that can be embedded over any specific set of $N$ points in the plane. More specifically, we bound the ratio between the number of spanning cycles (or perfect matchings) that can be embedded over a point set and the number of triangulations that can be embedded over it. The respective bounds are $O(1.8181^N)$ for cycles and $O(1.1067^N)$ for matchings. These imply a new upper bound of $O(54.543^N)$ on the number of crossing-free straight-edge spanning cycles that can be embedded over any specific set of $N$ points in the plane (improving upon the previous best upper bound $O(68.664^N)$). Our analysis is based on Kasteleyn's linear algebra technique.

Motivation & Objective

• To improve the upper bound on the number of crossing-free straight-edge spanning cycles (Hamiltonian cycles) that can be embedded over a set of N points in the plane.
• To analyze the relationship between the number of such cycles and the number of triangulations over the same point set.
• To apply Kasteleyn’s technique and edge-flipping methods to derive tighter combinatorial bounds.
• To establish a threshold-based bound that optimally combines two different analytical lemmas depending on the number of degree-3 interior vertices in a triangulation.
• To refine the overall asymptotic bound by leveraging structural properties of triangulations and separable edges.

Proposed method

• Uses Kasteleyn’s linear algebra technique for counting perfect matchings in planar graphs, adapted to bound the number of spanning cycles.
• Applies edge-flipping techniques from prior work to analyze the number of flippable edges in triangulations, which relates to cycle support and enumeration.
• Sorts triangulations by the number of interior vertices of degree 3 (v₃(T)) and applies different bounds based on this parameter.
• Employs two lemmas: one for triangulations with many degree-3 vertices (Lemma 5.1), another for those with fewer (Lemma 5.3), with a threshold at approximately 10.72% of N.
• Solves a complex equation involving exponential terms and parameters (x ≈ 1.17965, γ) to determine the optimal switching point between the two lemmas.
• Combines the resulting bounds with the known upper bound of tr(N) < 30^N on the number of triangulations to derive the final asymptotic result.

Experimental results

Research questions

• RQ1 What is the tightest possible upper bound on the number of crossing-free straight-edge Hamiltonian cycles over a set of N points in the plane?
• RQ2 How does the number of such cycles relate to the number of triangulations of the same point set?
• RQ3 Can Kasteleyn’s technique be effectively adapted to bound the number of spanning cycles in point sets with general position constraints?
• RQ4 At what threshold of degree-3 interior vertices in a triangulation does the optimal bound switch between two different analytical approaches?
• RQ5 Can edge-flipping and separable edge properties be used to refine combinatorial bounds on cycle enumeration in planar point sets?

Key findings

• The paper establishes a new upper bound of O(54.5430^N) on the maximum number of crossing-free straight-edge spanning cycles (simple polygonizations) over any set of N points in the plane.
• This improves upon the previous best-known upper bound of O(68.664^N), representing a significant tightening of the asymptotic estimate.
• The bound is derived by relating the number of spanning cycles to the number of triangulations via a factor of O(1.8181^N), which is the ratio between the number of cycles and triangulations.
• A threshold at approximately 10.72% of N degree-3 interior vertices determines the optimal use of two different analytical lemmas in the bound derivation.
• The analysis confirms that the combined use of Kasteleyn’s technique and triangulation-based edge-flipping yields a tighter bound than previous methods.
• The final bound is obtained by combining the refined cycle count with the known upper bound tr(N) < 30^N on the number of triangulations.