Long paths in hypercubes with conditional node-faults

Let F be a set of image faulty nodes in an n-cube image such that every node of image still has at least two fault-free neighbors. Then we show that image contains a path of length at least image (respectively, image) between any two nodes of odd (respectively, even) distance. Since the n-cube is bipartite, the path of length image (or image) 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 image faulty nodes are considered.