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 Fv be the set of faulty vertices in the n-dimensional enhanced hypercube Qn,k（1 ≤ k≤n- 1）.When ｜F_v｜ = 2,we showed that Qn,k-Fv contains a fault-free cycle of every even length from 4 to 2^n- 4 where n（n ≥ 3） and fc 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 — fc ＋ 2 to 2^n-3 where n（≥ 3） and fc have the different parity.Furthermore,when ｜Fv｜= fv ≤ n- 2,we proof that there exists the longest fault-free cycle,which is of even length 2^n- 2fv whether n（n 〉 3） and fe have the same parity or not;and there exists the longest fault-free cycle,which is of odd length 2^n-2fv- 1 in Qn,k — Fv where n（≥ 3） and fc have the different parity.