Edge-fault-tolerant edge-bipancyclicity of balanced hypercubes

The balanced hypercube, $BH_n$, is a variant of hypercube $Q_n$. R.X. Hao et al. $(2014)$ \cite{R.X.Hao} showed that there exists a fault-free Hamiltonian path between any two adjacent vertices in $BH_n$ with $(2n-2)$ faulty edges. D.Q. Cheng et al. $(2015)$ \cite{Dongqincheng2} proved that $BH_n$ is $6$-edge-bipancyclic after $(2n-3)$ faulty edges occur for all $n\ge2$. In this paper, we improve these two results by demonstrating that $BH_n$ is $6$-edge-bipancyclic even when there exist $(2n-2)$ faulty edges for all $n\ge2$. Our result is optimal with respect to the maximum number of tolerated edge faults.