In this paper, we study the enhanced hypercube, an attractive variant of the hypercube and obtained by adding some complementary edges from a hypercube, and focus on cycles embedding on the enhanced hypercube with faulty vertices. Let Fu be the set of faulty vertices in the n-dimensional enhanced hypercube Qn,k （n ≥ 3, 1 ≤ k 〈≤n - 1）. When IFvl = 2, we showed that Qn,k - Fv 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 2^n-3 where n （≥ 3） and k have the different parity. Furthermore, when ｜Fv｜ = fv ≤ n - 2, we prove that there exists the longest fault-free cycle, which is of even length 2^n - 2fv 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 - 2fv ＋ 1 in Qn,k - Fv where n （≥ 3） and k have the different parity.
For a fixed graph F,a graph G is F-saturated if it has no F as a subgraph,but does contain F after the addition of any new edge.The saturation number,sat(n,F),is the minimum number of edges of a graph in the set of all F-saturated graphs with order n.In this paper,we determine the saturation number sat(n,2P3∪tP2)and characterize the extremal graphs for n≥6t+8.
Let Qn,k （n 〉 3, 1 〈 k ≤ n - 1） be an n-dimensional enhanced hypercube which is an attractive variant of the hypercube and can be obtained by adding some complementary edges, fv and fe be the numbers of faulty vertices and faulty edges, respectively. In this paper, we give three main results. First, a fault-free path P[u, v] of length at least 2n - 2fv - 1 （respectively, 2n - 2fv - 2） can be embedded on Qn,k with fv ＋ f≤ n- 1 when dQn,k （u, v） is odd （respectively, dQ,~,k （u, v） is even）. Secondly, an Q,,k is （n - 2） edgefault-free hyper Hamiltonianaceable when n （ 3） and k have the same parity. Lastly, a fault-free cycle of length at least 2n - 2fv can be embedded on Qn,k with f~ 〈 n - 1 and fv＋f≤2n-4.
