Embedded paths and cycles in faulty hypercubes

An important task in the theory of hypercubes is to establish the maximum integer f n such that for every set ? of f vertices in the hypercube ${\mathcal {Q}}_{n},$ with 0?f?f n , there exists a cycle of length at least 2 n ?2f in the complement of ?. Until recently, exact values of f n were known only for n?4, and the best lower bound available for f n with n?5 was 2n?4. We prove that f 5=8 and obtain the lower bound f n ?3n?7 for all n?5. Our results and an example provided in the paper support the conjecture that $f_{n}={n\choose 2}-2$ for each n?4. New results regarding the existence of longest fault-free paths with prescribed ends are also proved.