Fault-free cycles passing through prescribed paths in hypercubes with faulty edges

An n -dimensional hypercube, or n -cube, denoted by Q n , is well known as bipartite and one of the most efficient networks for parallel computation. In this work, we consider the problem of cycles passing through prescribed paths in an n -dimensional hypercube with faulty edges. We obtain the following result: For n ≥ 3 and 2 ≤ h < n , let F be a subset of E ( Q n ) with | F | < n − h . Then, every fault-free path P with length h lies on a fault-free cycle in Q n − F of every even length from d to 2 n inclusive where d = 2 h if h > | F | + 1 and d = 2 h + 2 otherwise. The result is optimal.