Title:
Counting Hamiltonian cycles in planar triangulations
Counting Hamiltonian cycles in planar triangulations
Authors
Liu, Xiaonan
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Yu, Xingxing
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Abstract
Whitney showed that every planar triangulation without separating $3$-cycles is Hamiltonian. This result was extended to all $4$-connected planar graphs by Tutte. Hakimi, Schmeichel, and Thomassen showed the first lower bound $n/\log _2 n$ for the number of Hamiltonian cycles in every $n$-vertex $4$-connected planar triangulation and, in the same paper, they conjectured that this number is at least $2(n-2)(n-4)$, with equality if and only if $G$ is a double wheel. 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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Date Issued
2023-04-25
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Dissertation