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.