Vertex-fault-tolerant cycles embedding in balanced hypercubes

• For any edge e=(u,v)e=(u,v), there are 64(n-1)64(n-1) paths of length 7 joining u and v.
• For any edge e=(u,v)e=(u,v), there are 2(n-1)2(n-1) internal node-disjoint paths of length 7 joining u and v.
• For any fault-free edge e  , there exists a fault-free cycle of length 22n-2|Fv|22n-2|Fv| (|Fv|⩽n-2)|Fv|⩽n-2) containing e.
• Every fault-free edge lies on a fault-free cycle of every even length l  , where 4⩽l⩽22n-2|Fv|4⩽l⩽22n-2|Fv| (|Fv|⩽n-1)|Fv|⩽n-1).

The balanced hypercube is a new variant of the hypercube, which was proposed by Wu and Huang. Xu et al. (2007) proved that every edge of an n  -dimensional balanced hypercube BHnBHn lies on a cycle of every even length from 4 to 22n22n. In this paper, we consider the edge-bipancyclicity of BHnBHn with faulty vertices. Let FvFv be the set of faulty vertices in BHnBHn with |Fv|⩽n-1|Fv|⩽n-1. We show that every fault-free edge of BHn-FvBHn-Fv lies on a fault-free cycle of every even length from 4 to 22n-2|Fv|22n-2|Fv|, where n⩾1n⩾1. Our result improves the previous best result by Xu et al. in terms of fault-tolerant vertices.