Mutually Independent Hamiltonian Cycles in k-ary n-cubes when k is odd

The k-ary n-cubes, CA, is one of the most well-known interconnection networks in parallel computers. Let n >= 1 be an integer and k >= 3 be an odd integer. It has been shown that any Q(n)(k) is a 2n-regular, vertex symmetric and edge symmetric graph with a hamiltonian cycle. In this article, we prove that any k-ary n-cube contains 2n mutually independent hamiltonian cycles. More specifically, let v(i) is an element of V(Q(n)(k)) for 0 <= i <= vertical bar Q(n)(k)vertical bar - 1 and let < v(0),v(i),...,v(vertical bar Qnk vertical bar-1), v0 > be a hamiltonian cycle of Q(n)(k). We prove that Q(n)(k) contains 2n hamiltonian cycles of the form < v(0), v(1)(l) ,..., v(vertical bar Qnk vertical bar)(l)-1; v(0)> for 0 <= 1 <= 2n - 1, where v(i)(l) not equal v(i)(l)' whenever l not equal l'. The result is optimal since each vertex of Q(n)(k) has exactly 2n neighbors.