  The balanced hypercube $BH_{n}$, 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 $n\geq2$. In this paper, we show that there exists a fault-free
Hamiltonian cycle in $BH_{n}$ for $n\ge 2$ with $\left | F \right |\le 5n-7$ if
the degree of every vertex in $BH_{n}-F$ is at least two and there exists no
$f_{4}$-cycles in $BH_{n}-F$, which improves some known results.
