Enumerations of the Hamiltonian walks on a cubic sublattice

A massively parallel supercomputer was used to exhaustively enumerate all of the Hamiltonian walks for simple cubic sublattices of four different sizes (up to 3*4*4). The behaviour of the logarithm of the number of walks was found to be linear in the number of vertices in the lattice. The linear fit is shown to agree also with the asymptotic limit of the Flory mean field theoretical estimate. Thus, we suggest that the fit obtained yields the number of walks for any size fragment of the cubic lattice to logarithmic accuracy. The significance of this result to the validity of polymer models is also discussed.