Hamiltonian paths passing through matchings in hypercubes with faulty edges

Chen considered the existence of a Hamiltonian cycle containing a matching and avoiding some edges in an $n$-cube $Q_n$. In this paper, we considered the existence of a Hamiltonian path and obtained the following result. For $n\ge 4$, let $M$ be a matching of $Q_n$, and let $F$ be a set of edges in $Q_n-M$ with $|M\cup F|\le 2n-6$. Let $x$ and $y$ be two vertices of $Q_n$ with different parities satisfying $xy\notin M$. If all vertices in $Q_n-F$ have a degree of at least $2$, then there exists a Hamiltonian path joining $x$ and $y$ passing through $M$ in $Q_n-F$, with the exception of two cases: (1) there exist two neighbors $v$ and $t$ of $x$ (or $y$) satisfying $d_{Q_n-F}(v)=2$ and $xt\in M$ (or $yt\in M$); (2) there exists a path $xvuy$ of length 3 satisfying $d_{Q_n-F}(v)=2$ and $uy\in M$ or $d_{Q_n-F}(u)=2$ and $xv\in M$.