Every edge lies on cycles embedding in folded hypercubes with vertex-fault-tolerant

The folded hypercube is a well-known variation of hypercube structure and can be constructed from a hypercube by adding a link to every pair of vertices with complementary addresses. An n  -dimensional folded hypercube (FQnFQn for short) for any odd n is known to be bipartite. In this paper, let f   be a faulty vertex in FQnFQn. It has been shown that (1) Every edge of FQn−{f}FQn−{f} lies on a fault-free cycle of every even length l   with 4≤l≤2n−24≤l≤2n−2 where n≥3n≥3; (2) Every edge of FQn−{f}FQn−{f} lies on a fault-free cycle of every odd length l   with n+1≤l≤2n−1n+1≤l≤2n−1, where n≥2n≥2 is even. In terms of every edge lies on a fault-free cycle of every odd length in FQn−{f}FQn−{f}, our result improves the result of Cheng et al. (2013) where odd cycle length up to 2n−32n−3.