Long paths and cycles in faulty hypercubes: existence, optimality, complexity

A fault-free cycle in the n-dimensional hypercube Q n with f faulty vertices is long if it has length at least 2 n − 2 f . If all faulty vertices are from the same bipartite class of Q n , such length is the best possible. We prove a conjecture of Castañeda and Gotchev [N. Castañeda and I. S. Gotchev. Embedded paths and cycles in faulty hypercubes. J. Comb. Optim., 2009. doi:10.1007/s10878-008-9205-6.] asserting that f n = ( n 2 ) − 2 where f n for every set of at most f n faulty vertices, there exists a long fault-free cycle in Q n . Furthermore, we present several results on similar problems of long paths and long routings in faulty hypercubes and their complexity.