Hamiltonian paths and cycles passing through a prescribed path in hypercubes

Assume that P is any path in a bipartite graph G of length k with 2⩽k⩽h, G is said to be h-path bipancyclic if there exists a cycle C in G of every even length from 2k to |V(G)| such that P lies in C. In this paper, the following result is obtained: The n-dimensional hypercube Qn with n⩾3 is (2n−3)-path bipancyclic but is not (2n−2)-path bipancyclic, moreover, a path P of length k with 2⩽k⩽2n−3 lies in a cycle of length 2k−2 if and only if P contains two edges of the same dimension. In order to prove the above result we first show that any path of length at most 2n−1 is a subpath of a Hamiltonian path in Qn with n⩾2, moreover, the upper bound 2n−1 is sharp when n⩾4.