Long paths in hypercubes with a quadratic number of faults

A path between distinct vertices u and v of the n-dimensional hypercube image avoiding a given set of f faulty vertices is called long if its length is at least image. We present a function image such that if image then there is a long fault-free path between every pair of distinct vertices of the largest fault-free block of image. Moreover, the bound provided by image is asymptotically optimal. Furthermore, we show that assuming image, the existence of a long fault-free path between an arbitrary pair of vertices may be verified in polynomial time with respect to n and, if the path exists, its construction performed in linear time with respect to its length.