Various cycles embedding in faulty balanced hypercubes

•Consider the balanced hypercube BHnBHn with |Fe|≤2n-3|Fe|≤2n-3 faulty edges.•Prove that every edge of BHn-FeBHn-Fe lies on fault-free cycles of even lengths from 6 to 22n.•Prove that the lower limit of the length 6 is sharp.

Quite a lot of interconnection networks are served as the underlying topologies of large-scale multiprocessor systems. The hypercube is one of the most popular interconnection networks. In this paper we consider the balanced hypercube, which is a variant of the hypercube. Huang and Wu showed that the balanced hypercube has better properties than hypercube with the same number of links and processors. Let FeFe be the set of faulty edges in an n  -dimensional balanced hypercube BHnBHn, where n⩾2n⩾2. In this paper, we consider BHnBHn with |Fe|⩽2n-3|Fe|⩽2n-3 faulty edges and prove that every fault-free edge lies on a fault-free cycle of every even length from 6 to 22n22n in BHn-FeBHn-Fe. Furthermore, we prove that the lower limit of the length 6 is sharp by giving a counter example.