Hamiltonian paths on the Sierpinski gasket

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 $\ell >0$.

arXiv comment33 pages, 21 figures

References in corpus (1)

• Scaling of Hamiltonian walks on fractal lattices