Edge-bipancyclicity of a hypercube with faulty vertices and edges

A bipartite graph G=(V,E) is said to be bipancyclic if it contains a cycle of every even length from 4 to |V|. Furthermore, a bipancyclic G is said to be edge-bipancyclic if every edge of G lies on a cycle of every even length. Let F_v (respectively, F_e) be the set of faulty vertices (respectively, faulty edges) in an n-dimensional hypercube Q_n. In this paper, we show that every edge of Q_n-F_v-F_e lies on a cycle of every even length from 4 to 2ⁿ-2|F_v| even if |F_v|+|F_e|?n-2, where n?3. Since Q_n is bipartite of equal-size partite sets and is regular of vertex-degree n, both the number of faults tolerated and the length of a longest fault-free cycle obtained are worst-case optimal.