Fault-tolerance of balanced hypercubes with faulty vertices and faulty   edges
  Let $F_{v}$ (resp. $F_e$) be the set of faulty vertices (resp. faulty edges) in the $n$-dimensional balanced hypercube $BH_n$. Fault-tolerant Hamiltonian laceability in $BH_n$ with at most $2n-2$ faulty edges is obtained in [Inform. Sci. 300 (2015) 20--27]. The existence of edge-Hamiltonian cycles in $BH_n-F_e$ for $|F_e|\leq 2n-2$ are gotten in [Appl. Math. Comput. 244 (2014) 447--456]. Up to now, almost all results about fault-tolerance in $BH_n$ with only faulty vertices or only faulty edges. In this paper, we consider fault-tolerant cycle embedding of $BH_n$ with both faulty vertices and faulty edges, and prove that there exists a fault-free cycle of length $2^{2n}-2|F_v|$ in $BH_n$ with $|F_v|+|F_e|\leq 2n-2$ and $|F_v|\leq n-1$ for $n\geq 2$. Since $BH_n$ is a bipartite graph with two partite sets of equal size, the cycle of a length $2^{2n}-2|F_v|$ is the longest in the worst-case. 
Mei-Mei Gu, Rong-Xia Hao
math.CO