Matchings extend to Hamiltonian cycles in k-ary n-cubes

The k-ary n-cube is one of the most popular interconnection networks for parallel and distributed systems. Given an edge set in the k-ary n-cube, which conditions guarantee the existence of a Hamiltonian cycle in the k-ary n  -cube containing the edge set? In this paper, we prove for n⩾2n⩾2 and k⩾3k⩾3 that every matching having at most 3n-83n-8 edges is contained in a Hamiltonian cycle in the k-ary n-cube. Also, we present an example to show that the analogous conclusion does not hold for perfect matchings.