Counting cycles in planar triangulations

TL;DR

This paper studies the minimum and maximum number of cycles of various lengths in planar triangulations, revealing bounds related to the triangulation's dual radius and constructing examples with many cycles.

Contribution

It establishes new lower bounds on cycle counts in planar triangulations and constructs examples with many cycles, advancing understanding of cycle distribution.

Findings

Minimum cycle count is linear in n for certain lengths.

Existence of planar Hamiltonian triangulations with O(n) cycles for many lengths.

Planar 4-connected triangulations contain linear or quadratic numbers of cycles depending on conditions.

Abstract

We investigate the minimum number of cycles of specified lengths in planar $n$-vertex triangulations $G$. It is proven that this number is $\Omega(n)$ for any cycle length at most $3 + \max \{ {\rm rad}(G^*), \lceil (\frac{n-3}{2})^{\log_32} \rceil \}$, where ${\rm rad}(G^*)$ denotes the radius of the triangulation's dual, which is at least logarithmic but can be linear in the order of the triangulation. We also show that there exist planar hamiltonian $n$-vertex triangulations containing $O(n)$ many $k$-cycles for any $k \in \{ \lceil n - \sqrt[5]{n} \rceil, \ldots, n \}$. Furthermore, we prove that planar 4-connected $n$-vertex triangulations contain $\Omega(n)$ many $k$-cycles for every $k \in \{ 3, \ldots, n \}$, and that, under certain additional conditions, they contain $\Omega(n^2)$ $k$-cycles for many values of $k$, including $n$.