Long paths and cycles in hypercubes with faulty vertices

A fault-free path in the image-dimensional hypercube image with image faulty vertices is said to be long if it has length at least image. Similarly, a fault-free cycle in image is long if it has length at least image. If all faulty vertices are from the same bipartite class of image, such length is the best possible. We show that for every set of at most image faulty vertices in image and every two fault-free vertices image and image satisfying a simple necessary condition on neighbors of image and image, there exists a long fault-free path between image and image. This number of faulty vertices is tight and improves the previously known results. Furthermore, we show for every set of at most image faulty vertices in image where image that image has a long fault-free cycle. This is a first quadratic bound, which is known to be asymptotically optimal.