Fault-Tolerant Cycle Embedding in Balanced Hypercubes with Faulty Vertices and Faulty Edges

Abstract

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 | ≤ 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 | ≤ 2n − 2 and |F v | ≤ n − 1 for n ≥ 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.

Constructing Two Completely Independent Spanning Trees in Balanced Hypercubes

“…A particular property of the balanced hypercube is that each processor has a backup processor that shares the same neighborhood. For more results about the balanced hypercubes, please refer to [4], [8]- [11].…”