Hamiltonian cycles of balanced hypercube with more faulty edges

The balanced hypercube $B{H}_n$, a variant of the hypercube, is a novel interconnection network for massive parallel systems. It is known that the balanced hypercube remains Hamiltonian after deleting at most $4n-5$ faulty edges if each vertex is incident with at least two edges in the resulting graph for all $n\ge 2$. In this paper, we show that there exists a fault-free Hamiltonian cycle in $B{H}_n$ for $n\ge 2$ with $\mid F\mid \le 5n-7$ if the degree of every vertex in $B{H}_n-F$ is at least two and there exists no $f_4$-cycles in $B{H}_n-F$, which improves some known results.