Fault-tolerant edge-bipancyclicity of faulty hypercubes under the conditional-fault model

It is well-known that the n-dimensional hypercube Qn is one of the most versatile and efficient interconnection network architecture yet discovered for building massively parallel or distributed systems. Let F be the faulty set of Qn and let fv, fe be the numbers of faulty vertices and faulty edges in F, respectively. An edge e=(x,y) is said to be free if e, x, y are not in F, and a cycle is said to be fault-free if there is no faulty vertex or faulty edge on the cycle. In this paper, we prove that each free edge (x, y) in Qn for n ≥ 3 lies on a fault-free cycle of any even length from 6 to 2n−2fv if fv+fe≤2n−5,fe≤n−2 and both x and y are incident to at least two free edges. This result confirms a conjecture reported in the literature.