Many algorithms in parallel systems can be easily solved if we can generate a Hamiltonian cycle on the underly network. Finding Hamiltonian cycle is a well known NP-complete problem. For specific instances of regular graphs, such as Torus and Gaussian network, one can easily find Hamiltonian cycles. In this thesis, we present a recurrence function that can generate 2[superscript r] ≥ 1 independent Gray codes from Z[supserscript n][subscript k] where 2[superscript r] ≤ n < 2[superscript r+1]. Such independent Gray codes corresponds to edge disjoint Hamiltonian cycles on the Torus graph T[supserscript n][subscript k] and multidimensional Gaussian network Gα[superscript ⌊n/2⌋], for 1 ≤ 2[superscript r] ≤ n < 2[superscript r+1].