Odd cycles embedding on folded hypercubes with conditional faulty edges

Let FFe be the set of |FFe|⩽2n-5 faulty edges in an n-dimensional folded hypercube FQn such that each vertex of FQn is incident to at least two fault-free edges, where n⩾4 and n is even. Under this assumption, we show that every fault-free edge of FQn lies on a fault-free cycle of every odd length from n+1 to 2n-1. In terms of the number of tolerant faulty edges and embedding odd cycles in FQn, our result improves not only the result in Xu and Ma (2006) where |FFe|=0, but also the previous best result gotten by Xu et al. (2006) where |FFe|⩽n-1.