Prescribed matchings extend to Hamiltonian cycles in hypercubes with faulty edges

Ruskey and Savage asked the following question: for n ≥ 2 , does every matching in Q n extend to a Hamiltonian cycle in Q n ? Fink showed that the answer is yes for every perfect matching, thereby proving Kreweras’ conjecture. In this paper we consider the question in hypercubes with faulty edges. We show for n ≥ 2 that every matching M of at most 2 n − 1 edges extends to a Hamiltonian cycle in Q n . Moreover, we prove that when n ≥ 4 and M is nonempty this conclusion still holds even if Q n has at most n − 1 − ⌈ | M | 2 ⌉ faulty edges, with one exception.