A kind of matchings extend to Hamiltonian cycles in hypercubes

Ruskey and Savage asked the following question: Does every matching in Q(n) for n >= 2 extend to a Hamiltonian cycle of Q(n)? Kreweras conjectured that every perfect matching of Q(n) for n >= 2 can be extended to a Hamiltonian cycle of Q(n). Fink confirmed the conjecture. An edge in Qn is an edge of direction i if its endpoints differ in the ith position. So all the edges of Q(n) can be divided into n directions, i.e., edges of direction 1, & mldr;, edges of direction n. The set of all edges of direction i of Q(n) is denoted by E-i. In this paper, we obtain the following result. For n >= 6, let M be a matching in Q(n) with |M| < 10 x 2(n-5). If M contains edges in at most 5 directions, then M can be extended to a Hamiltonian cycle of Q(n).