Extending pairings to Hamiltonian cycles

Recently J. Fink proved that every $1$-factor of the complete graph on the vertex set of the hypercube $Q_n$ can be extended to a cycle by adding some edges of this hypercube. We prove that, for $n\ge4$, one can remove some edges of $Q_n$ so that the resulting graph still has this property. Also we give upper and lower bounds on the minimum number of edges of a $2n$-vertex graph having this property.