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 com-plementary 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+fe ≤ n−1 when dQn,k (u, v) is odd (respectively, dQn,k (u, v) is even). Secondly, an Qn,k is (n − 2) edge-fault-free hyper Hamiltonian-laceable 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 fe ≤ n − 1 and fv + fe ≤ 2n − 4.
