Various cycles embedding in faulty balanced hypercubes

Consider the balanced hypercube BH n with | F e | ? 2 n - 3 faulty edges.Prove that every edge of BH n - F e 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 F e be the set of faulty edges in an n-dimensional balanced hypercube BH n , where n ? 2 . In this paper, we consider BH n with | F e | ≤ 2 n - 3 faulty edges and prove that every fault-free edge lies on a fault-free cycle of every even length from 6 to 2 2 n in BH n - F e . Furthermore, we prove that the lower limit of the length 6 is sharp by giving a counter example.

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