Hamiltonian cycles of balanced hypercube with disjoint faulty edges

The balanced hypercube $B{H}_n$, a variant of the hypercube, is a novel interconnection network topology for massive parallel systems. It is showed in [Theor. Comput. Sci. 947 (2023) 113708] that for any edge subset F of $B{H}_n$ there exists a fault-free Hamiltonian cycle in $B{H}_n-F$ for $n\ge 2$ with $\left|F\right|\le 5n-7$ if the degree of every vertex in $B{H}_n-F$ is at least two and there exist no $f_4$-cycles in $B{H}_n-F$. In this paper, we consider the existence of Hamiltonian cycles of $B{H}_n$ when F is a matching (a set of disjoint edges), and show that each edge $e\notin F$ lies on a fault-free Hamiltonian cycle of $B{H}_n-F$ with $n\ge 2$. The number of faulty edges in F can be up to $2^{2n-1}$, which is exponential to the dimension n.