Extending the Size of Matchings to Hamiltonian Cycles in Hypercubes

The hypercube [Formula: see text] is a well-known interconnection network topology due to its symmetry, regularity, and recursive structure. A longstanding open problem posed by Ruskey and Savage asks whether every matching in [Formula: see text], for [Formula: see text], can be extended to a Hamiltonian cycle. Despite over three decades, this problem remains unresolved. However, this conjecture holds for certain special classes of matchings, such as perfect matchings, linear matchings, and some kinds of matchings. It is important to note that not every matching in a hypercube can be extended to a perfect matching. In this study, we contribute to this problem by proving that any matching in the [Formula: see text]-dimensional hypercube [Formula: see text] with at most [Formula: see text] edges can be extended to a Hamiltonian cycle, for [Formula: see text].