Long paths in hypercubes with conditional node-faults

Let F   be a set of f⩽2n-5f⩽2n-5 faulty nodes in an n  -cube QnQn such that every node of QnQn still has at least two fault-free neighbors. Then we show that Qn-FQn-F contains a path of length at least 2n-2f-12n-2f-1 (respectively, 2n-2f-22n-2f-2) between any two nodes of odd (respectively, even) distance. Since the n  -cube is bipartite, the path of length 2n-2f-12n-2f-1 (or 2n-2f-22n-2f-2) turns out to be the longest if all faulty nodes belong to the same partite set. As a contribution, our study improves upon the previous result presented by [J.-S. Fu, Longest fault-free paths in hypercubes with vertex faults, Information Sciences 176 (2006) 759–771] where only n-2n-2 faulty nodes are considered.