Hamiltonian paths extending a set of matchings in hypercubes

The hypercube \ (Qₙ \) contains a Hamiltonian path joining \ (x \) and \ (y \) (where x and y from the opposite partite set) containing \ (P \) if and only if the induced subgraph of \ (P \) is a linear forest, where none of these paths have \ (x \) or \ (y \) as internal vertices nor both as endpoints. Dvořák and Gregor answered a problem posed by Caha and Koubek and proved that for every \ (n 5 \), there exist vertices \ (x \) and \ (y \) with a set of \ (2n - 4 \) edges in \ (Qₙ \) that extend to the Hamiltonian path joining \ (x \) and \ (y \). This paper examines the Hamiltonian properties of hypercubes with a matching set. Let consider the hypercube \ (Qₙ \), for \ (n 5 \) and a set of matching \ (M \) such that \ (|M| 3n - 13 \). We prove a Hamiltonian path exists joining two vertices x and y in \ (Qₙ \) from opposite partite sets containing M.