Fault-tolerant cycles embedding in hypercubes with faulty edges

Let Qn be an n-dimensional hypercube with fe⩽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 2n in Qn. 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.