Long cycles in hypercubes with optimal number of faulty vertices

Let f(n) be the maximum integer such that for every set F of at most f(n) vertices of the hypercube Qn, there exists a cycle of length at least 2n−2|F| in Qn−F. Castañeda and Gotchev conjectured that $f(n)=\binom{n}{2}-2$. We prove this conjecture. We also prove that for every set F of at most (n2+n−4)/4 vertices of Qn, there exists a path of length at least 2n−2|F|−2 in Qn−F between any two vertices such that each of them has at most 3 neighbors in F. We introduce a new technique of potentials which could be of independent interest.