Hamiltonian walks on the Sierpinski gasket

Abstract

We derive the exact number of Hamiltonian walks 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$3(23)3n−13×(52×72×172212×35×13)(16)n. We also obtain the number of Hamiltonian walks with one end at a specific 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$3(23)3n−13×(7×1724×33)4n. The distribution of Hamiltonian walks on SG(n) with one end at a specific outmost vertex and the other at an arbitrary vertex of SG(n) is investigated. We rigorously prove that the exponent for the mean ℓ displacement between the two end vertices of such Hamiltonian walks on SG(n) is ℓln 2/ln 3 for ℓ > 0.