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We study Hamiltonian walks (HWs) on the family of three--dimensional modifiedSierpinski gasket fractals, as a model for compact polymers in nonhomogeneousmedia in three dimensions. Each member of this fractal family is labeled withan integer $b\geq 2$. We apply an exact recursive method which allows forexplicit enumeration of extremely long Hamiltonian walks of different types:closed and open, with end-points anywhere in the lattice, or with one or bothends fixed at the corner sites, as well as some Hamiltonian conformationsconsisting of two or three strands. Analyzing large sets of data obtained for$b=2,3$ and 4, we find that numbers $Z_N$ of Hamiltonian walks, on fractallattice with $N$ sites, for $N\gg 1$ behave as $Z_N\sim \omega^N\mu^{N^\sigma}$. The leading term $\omega^N$ is characterized by the value ofthe connectivity constant $\omega>1$, which depends on $b$, but not on the typeof HW. In contrast to that, the stretched exponential term $\mu^{N^\sigma}$depends on the type of HW through constant $\mu<1$, whereas exponent $\sigma$is determined by $b$ alone. For larger $b$ values, using some general featuresof the applied recursive relations, without explicit enumeration of HWs, weargue that asymptotical behavior of $Z_N$ should be the same, with $\sigma=\ln3/\ln[b(b+1)(b+2)/6]$, valid for all $b>2$. This differs from the formulaeobtained recently for Hamiltonian walks on other fractal lattices, as well asfrom the formula expected for homogeneous lattices. We discuss the possibleorigin and implications of such result.

A model of compact polymers on a family of three-dimensional fractal lattices