Cycles embedding in balanced hypercubes with faulty edges and vertices

Wu and Huang proposed a new variation of the hypercube, named balanced hypercube, which possesses many good properties such as bipanconnectivity, edge-bipancyclicity, Hamiltonian laceability, hyper Hamiltonian laceability. In this paper, we consider n-dimensional balanced hypercube with |Fe| faulty edges and |Fv| faulty vertices. We prove that if |Fv|+|Fe|≤n−1, then every fault-free edge of BHn lies on a fault-free cycle of every even length from 6 to 22n−2|Fv|, where n≥2; and if |Fv|+|Fe|≤2n−3, then there is a fault-free cycle of every even length from 6 to 22n−2|Fv| in BHn, where n≥2. Furthermore, we propose the distance between vertex-disjoint edge e and cycle C, i.e., d(e,C)=min{d(e,e′)|e′∈E(C)}, where d(e,e′)=min{d(u,x),d(u,y),d(v,x),d(v,y)|(u,v)=e,(x,y)=e′}.