On the Enhanced Hyper-hamiltonian Laceability of Hypercubes

A bipartite graph is hamiltonian laceable if there exists a hamiltonian path between any two vertices that are in different partite sets. A hamiltonian laceable graph G is said to be hyper-hamiltonian laceable if, for any vertex v of G, there exists a hamiltonian path of G - {v} joining any two vertices that are located in the same partite set different from that of v. In this paper, we further improve the hyper-hamiltonian laceability of hypercubes by showing that, for any two vertices x, y from one partite set of Q(n), n >= 4, and any vertex w from the other partite set, there exists a hamiltonian path H of Q(n) - {w} joining x to y such that d(H)(X, Z) = l for any vertex z is an element of V(Q(n)) - {x, y, w} and for every integer l satisfying both d(Qn) (x, z) <= l <= 2(n) - 2 - d(Qn) (z, y) and 2 vertical bar(l - d(Qn) (x, z)). As a consequence, many attractive properties of hypercubes follow directly from our result.