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arXiv · 2105.07551

Counting Hamiltonian cycles in planar triangulations

Xiaonan Liu·Zhiyu Wang·Xingxing Yu
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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.

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Xiaonan Liu, Zhiyu Wang, Xingxing Yu. 2021-05-17. Counting Hamiltonian cycles in planar triangulations. https://arxiv.org/abs/2105.07551

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