Long paths in hypercubes with a quadratic number of faults

A path between distinct vertices u and v of the n  -dimensional hypercube QnQn avoiding a given set of f   faulty vertices is called long if its length is at least 2n-2f-22n-2f-2. We present a function ϕ(n)=Θ(n2)ϕ(n)=Θ(n2) such that if f⩽ϕ(n)f⩽ϕ(n) then there is a long fault-free path between every pair of distinct vertices of the largest fault-free block of QnQn. Moreover, the bound provided by ϕ(n)ϕ(n) is asymptotically optimal. Furthermore, we show that assuming f⩽ϕ(n)f⩽ϕ(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.