Hamiltonian paths extending a set of matchings in hypercubes

The hypercube $Q_n$ 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. Dvo\v{r}\'ak and Gregor answered a problem posed by Caha and Koubek and proved that for every $n \geq 5$, there exist vertices $x$ and $y$ with a set of $2n - 4$ edges in $Q_n$ 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 $Q_n$, for $n \geq 5$ and a set of matching $M$ such that $|M| \leq 3n - 13$. We prove a Hamiltonian path exists joining two vertices $x$ and $y$ in $Q_n$ from opposite partite sets containing $M$.