Long paths and cycles in hypercubes with faulty vertices

A fault-free path in the n-dimensional hypercube Q"n with f faulty vertices is said to be long if it has length at least 2^n-2f-2. Similarly, a fault-free cycle in Q"n is long if it has length at least 2^n-2f. If all faulty vertices are from the same bipartite class of Q"n, such length is the best possible. We show that for every set of at most 2n-4 faulty vertices in Q"n and every two fault-free vertices u and v satisfying a simple necessary condition on neighbors of u and v, there exists a long fault-free path between u and v. This number of faulty vertices is tight and improves the previously known results. Furthermore, we show for every set of at most n^2/10+n/2+1 faulty vertices in Q"n where n>=15 that Q"n has a long fault-free cycle. This is a first quadratic bound, which is known to be asymptotically optimal.