Edge-bipancyclicity of a hypercube with faulty vertices and edges

A bipartite graph G=(V,E)G=(V,E) is said to be bipancyclic   if it contains a cycle of every even length from 4 to |V||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 FvFv (respectively, FeFe) be the set of faulty vertices (respectively, faulty edges) in an n  -dimensional hypercube QnQn. In this paper, we show that every edge of Qn-Fv-FeQn-Fv-Fe lies on a cycle of every even length from 4 to 2n-2|Fv|2n-2|Fv| even if |Fv|+|Fe|⩽n-2|Fv|+|Fe|⩽n-2, where n⩾3n⩾3. Since QnQn 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.