The Panpositionable Hamiltonicity of k-ary n-cubes

The hypercube Qn is one of the most well-known and popular interconnection networks and the k-ary n-cubeQn is an enlarged family from Qn that keeps many pleasing properties from hypercubes. In this article, we study the panpositionable hamiltonicity of Qn for k ≥ 3 and n ≥ 2. Let x, y of V (Qn) be two arbitrary vertices and C be a hamiltonian cycle of Qn. We use dC(x, y) to denote the distance between x and y on the hamiltonian cycle C. Define l as an integer satisfying d(x, y) ≤ l ≤ 1 2 |V (Qn)|. We prove the followings: • When k = 3 and n ≥ 2, there exists a hamiltonian cycle C of Qn such that dC(x, y) = l. • When k ≥ 5 is odd and n ≥ 2, we request that l / ∈ S where S is a set of specific integers. Then there exists a hamiltonian cycle C of Qn such that dC(x, y) = l. • When k ≥ 4 is even and n ≥ 2, we request l− d(x, y) to be even. Then there exists a hamiltonian cycle C of Qn such that dC(x, y) = l. The result is optimal since the restrictions on l is due to the structure of Qn by definition.