Vertex-Fault-Tolerant Cycles Embedding on Enhanced Hypercube Networks

In this paper, we focus on the vertex-fault-tolerant cycles embedding on enhanced hypercube, which is an attractive variant of hypercube and is obtained by adding some complementary edges from hypercube. Let F-v be the set of faulty vertices in the n-dimensional enhanced hypercube Q(n,k) (1 <= k <= n-1). When vertical bar F-v vertical bar = 2, we showed that Q(n,k) - F-v contains a fault-free cycle of every even length from 4 to 2(n) - 4 where n (n >= 3) and k have the same parity; and contains a fault-free cycle of every even length from 4 to 2(n) - 4, simultaneously, contains a cycle of every odd length from n - k + 2 to 2(n) - 3 where n(>= 3) and k have the different parity. Furthermore, when vertical bar F-v vertical bar = f(v) <= n - 2, we proof that there exists the longest fault-free cycle, which is of even length 2(n) - 2f(v) whether n(n >= 3) and k have the same parity or not; and there exists the longest fault-free cycle, which is of odd length 2(n) - 2f(v) - 1 in Q(n,k) - F-v where n(>= 3) and k have the different parity.