Mutually independent Hamiltonian cycles in k-ary n-cubes when k is even

The k-ary n-cubes, Qn, 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 Qn 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 vi ∈ V (Qn) for 0 ≤ i ≤ |Qn| − 1 and let 〈v0, v1, . . . , v|Qkn|−1, v0〉 be a hamiltonian cycle of Qn. We prove that Qn contains 2n hamiltonian cycles of the form 〈v0, vl 1, . . . , vl |Qkn|−1 v0〉 for 0 ≤ l ≤ 2n − 1, where v l i = vl ′ i whenever l = l′. The result is optimal since each vertex of Qn has exactly 2n neighbors. Key–Words: k-ary n-cubes, hamiltonian cycle, mutually independent, hypercubes.