A type of perfect matchings extend to hamiltonian cycles in k-ary n-cubes

Abstract

Kreweras conjectured that every perfect matching in a hypercube Qn for n≥2 extends to a hamiltonian cycle of Qn. Fink confirmed the conjecture to be true. The k-ary n-cube Qnk is a generalization of the hypercube. However, the analogous result does not necessarily hold for Qnk. We can find a perfect matching in Q26 which is not contained in any hamiltonian cycle of Q26. In this paper, we investigate the existence of a hamiltonian cycle passing through a perfect matching in Qnk. For an integer n≥2 and an even integer k≥6, we prove that every perfect matching in Qnk consisting of edges in the same dimension can be extended to a hamiltonian cycle of Qnk.