Vertex-fault-tolerant cycles embedding on enhanced hypercube networks

Abstract

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 |F _v | = 2, we showed that Q _n,k − F _v contains a fault-free cycle of every even length from 4 to 2ⁿ −4 where n (n ≥ 3) and k have the same parity; and contains a fault-free cycle of every even length from 4 to 2ⁿ − 4, simultaneously, contains a cycle of every odd length from n − k + 2 to 2ⁿ − 3 where n(≥ 3) and k have the different parity. Furthermore, when |F ^v| = f ^v ≤ n − 2, we proof that there exists the longest fault-free cycle, which is of even length 2ⁿ − 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 2ⁿ − 2f ^v − 1 in Q ^n,k − F ^v where n(≥ 3) and k have the different parity.