Long paths in hypercubes with a quadratic number of faults

A path between distinct vertices u and v of the n-dimensional hypercube Q_n avoiding a given set of f faulty vertices is called long if its length is at least 2ⁿ-2f-2. We present a function (n)=Θ(n²) such that if f(n) then there is a long fault-free path between every pair of distinct vertices of the largest fault-free block of Q_n. Moreover, the bound provided by (n) is asymptotically optimal. Furthermore, we show that assuming f(n), 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.