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 hyperhamiltonian 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 Qn, n ≥ 4, and any vertex w from the other partite set, there exists a hamiltonian path H of Qn − {w} joining x to y such that dH(x, z) = l for any vertex z ∈ V (Qn)− {x,y,w} and for every integer l satisfying both dQn(x, z) ≤ l ≤ 2 − 2 − dQn(z,y) and 2|(l − dQn(x, z)). As a consequence, many attractive properties of hypercubes follow directly from our result. Keywords— Path embedding, Hamiltonian laceable, Hyperhamiltonian laceable, Interconnection network, Hypercube