Hamiltonian paths on the Sierpinski gasket

TL;DR

This paper provides exact counts and asymptotic formulas for Hamiltonian paths on the Sierpinski gasket, analyzing their distribution and displacement properties with rigorous mathematical proofs.

Contribution

It derives exact and asymptotic counts of Hamiltonian paths on SG(n), including those with specific end vertices, and proves properties of their displacement behavior.

Findings

Exact number of Hamiltonian paths on SG(n)

Asymptotic formulas for path counts

Displacement exponent for paths on SG(n)

Abstract

We derive exactly the number of Hamiltonian paths H(n) on the two dimensional Sierpinski gasket SG(n) at stage $n$, whose asymptotic behavior is given by $\frac{\sqrt{3}(2\sqrt{3})^{3^{n-1}}}{3} \times (\frac{5^2 \times 7^2 \times 17^2}{2^{12} \times 3^5 \times 13})(16)^n$. We also obtain the number of Hamiltonian paths with one end at a certain outmost vertex of SG(n), with asymptotic behavior $\frac {\sqrt{3}(2\sqrt{3})^{3^{n-1}}}{3} \times (\frac {7 \times 17}{2^4 \times 3^3})4^n$. The distribution of Hamiltonian paths on SG(n) with one end at a certain outmost vertex and the other end at an arbitrary vertex of SG(n) is investigated. We rigorously prove that the exponent for the mean $\ell$ displacement between the two end vertices of such Hamiltonian paths on SG(n) is $\ell \log 2 / \log 3$ for $\ell>0$.