Fault-tolerant cycles embedding in hypercubes with faulty edges

Let Q(n) be an n-dimensional hypercube with f(e) <= 3n - 8 faulty edges and n >= 5. In this paper, we consider the faulty hypercube under the following two additional conditions: (1) each vertex is incident to at least two fault-free edges, and (2) every 4-cycle does not have any pair of non-adjacent vertices whose degrees are both two after removing the faulty edges. We prove that there exists a fault-free cycle of every even length from 4 to 2(n) in Q(n). Our result improves the result by Liu and Wang (2014) in terms of the lengths of embedding cycles, where under the same conditions, a fault-free Hamiltonian cycle was constructed. (C) 2014 Elsevier Inc. All rights reserved.