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The architecture of an interconnection network is usually represented by a graph, and a graph G is bipancyclic if it contains a cycle for every even length from 4 to |V(G)|. In this article, we analyze the conditional edge-fault-tolerant properties of an enhanced hypercube, which is an attractive variant of a hypercube that can be obtained by adding some complementary edges. For any n-dimensional enhanced hypercube with at most (2n-3) faulty edges in which each vertex is incident with at least two fault-free edges, we showed that there exists a fault-free cycle for every even length from 4 to 2n when n (n ≥ 3) and k have the same parity. We also show that a fault-free cycle for every odd length exists from n-k+2 to 2n-1 when n (n ≥ 2) and k have the different parity.