Embedding even cycles on folded hypercubes with conditional faulty edges

•Conditional fault means each node is incident to at least two fault-free edges.•Consider the folded hypercube FQnFQn with |FFe||FFe| faulty edges under the conditional fault.•Prove that every edge of FQn−FFeFQn−FFe lies on cycles of even lengths from 6 to 2n2n when |FFe|≤2n−4|FFe|≤2n−4, where n≥5n≥5.

Let FFeFFe be the set of |FFe|≤2n−4|FFe|≤2n−4 faulty edges in an n  -dimensional folded hypercube FQnFQn such that each vertex in FQnFQn is incident to at least two fault-free edges. Under this assumption, we show that every edge of FQn−FFeFQn−FFe lies on a fault-free cycle of every even length from 6 to 2n2n, where n≥5n≥5.