Long paths and cycles in hypercubes with faulty vertices

Abstract

A fault-free path in the n -dimensional hypercube Q n with f faulty vertices is said to be long if it has length at least 2 n - 2 f - 2 . Similarly, a fault-free cycle in Q n 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 show that for every set of at most 2 n - 4 faulty vertices in Q n and every two fault-free vertices u and v satisfying a simple necessary condition on neighbors of u and v , there exists a long fault-free path between u and v . This number of faulty vertices is tight and improves the previously known results. Furthermore, we show for every set of at most n 2 / 10 + n / 2 + 1 faulty vertices in Q n where n ⩾ 15 that Q n has a long fault-free cycle. This is a first quadratic bound, which is known to be asymptotically optimal.