Ruskey–Savage conjecture on extending hypercube matchings to Hamilton cycles

The $d$-dimensional hypercube $Q_d$ has vertex set consisting of all subsets of $[d]:=\{1,\ldots,d\}$, with edges between sets that differ in a single element. A matching is a set of pairwise disjoint edges. For $d\geq 2$, a Hamilton cycle is a cycle that visits every vertex of $Q_d$ exactly once. Ruskey–Savage conjecture. Every matching of $Q_d$, $d\geq 2$, can be extended to a Hamilton cycle. The conjecture asks whether arbitrary prescribed disjoint edges in a hypercube can be incorporated into a spanning cycle. The paper states a resolved result proving the natural relaxation that every matching extends to a cycle visiting at least a $2/3$-fraction of the vertices, while the full Hamilton-cycle assertion is the conjecture under discussion.