Towards a problem of Ruskey and Savage on matching extendability

Does every matching in the n-dimensional hypercube Qn extend to a Hamiltonian cycle? This question was raised by Ruskey and Savage in 1993 and even though a positive answer in known in some special cases, the problem still remains open in general. In this paper we present recent results on extendability of matchings in hypercubes to Hamiltonian cycles and paths as well as on the computational complexity of these problems, motivated by the Ruskey-Savage question. Moreover, we verify the conjecture of Vandenbussche and West saying that every matching in Qn, n≥2, extends to a 2-factor.