Counting Hamiltonian cycles in planar triangulations

TL;DR

This paper proves that 4-connected planar triangulations have at least quadratic numbers of Hamiltonian cycles, and under certain conditions, exponentially many, advancing understanding of cycle counts in such graphs.

Contribution

It establishes lower bounds on the number of Hamiltonian cycles in 4-connected planar triangulations, including exponential bounds under specific structural conditions.

Findings

Every 4-connected planar triangulation has at least Ω(n^2) Hamiltonian cycles.

If vertices of degree 4 are sufficiently spaced, the number of Hamiltonian cycles is at least 2^{Ω(n^{1/4})}.

Provides new bounds improving previous conjectures on Hamiltonian cycles in such graphs.

Abstract

Hakimi, Schmeichel, and Thomassen in 1979 conjectured that every $4$-connected planar triangulation $G$ on $n$ vertices has at least $2(n-2)(n-4)$ Hamiltonian cycles, with equality if and only if $G$ is a double wheel. In this paper, we show that every $4$-connected planar triangulation on $n$ vertices has $\Omega(n^2)$ Hamiltonian cycles. Moreover, we show that if $G$ is a $4$-connected planar triangulation on $n$ vertices and the distance between any two vertices of degree $4$ in $G$ is at least $3$, then $G$ has $2^{\Omega(n^{1/4})}$ Hamiltonian cycles.