Generalized Hypercubes: Edge-Disjoint Hamiltonian Cycles and Gray Codes

Some new classes of Hamming metric Gray codes over Z
_p
ⁿ
, where p is a prime and n is an integer power of 2, are described; then, how these Gray codes can be used to generate the maximum number of edge-disjoint Hamiltonian cycles in an n-dimensional generalized hypercube (GHC), Q
_p
ⁿ
, is shown. For Q
_p
ⁿ
, the number of edge-disjoint Hamiltonian cycles generated using these methods is n(p -1)/2 which is the maximum possible since the degree of each node in Q
_p
ⁿ
is n(p - 1). In addition, for any integers p and n, p not necessarily a prime and n not necessarily a power of 2, how to generate the maximum number of edge-disjoint Hamiltonian cycles in Qnp is also described.