Cycles embedding in folded hypercubes under the conditional fault model

A faulty network G is under the conditional fault model, i.e., every fault-free vertex of G is incident to at least two fault-free edges. Let FFv and FFe be the set of faulty vertices and faulty edges in FQn, respectively. In this paper, we consider FQn under the conditional fault model and prove that if |FFv|+|FFe|≤2n−4 and n≥3, then FQn−FFv−FFe contains a fault-free cycle of every even length from 4 to 2n−2|FFv|; if |FFv|+|FFe|≤2n−5 and n≥4 is even, then FQn−FFv−FFe contains a fault-free cycle of every odd length from n+1 to 2n−2|FFv|−1.