Matchings Extend to Perfect Matchings on Hypercube Networks

In this work, we investigate in the problem of perfect matchings with prescribed matchings in the n-dimensional hypercube network Qn. We obtain the following contributions: For any arbitrary matching with at most n − 1 edges, it can be extended to a perfect matching of Qn for n ≥ 1. Furthermore, for any arbitrary non-forbidden matching with n edges, it also can be extended to a perfect matching of Qn for n ≥ 1. It is shown by J. Fink in 2007 that any arbitrary perfect matching of the n-dimensional hypercube Qn, n ≥ 2, can be extended to a Hamiltonian cycle. Therefore, it leads to a further result that for any arbitrary non-forbidden matching with at most n edges, it can be extended to a Hamiltonian cycle of Qn for n ≥ 2.