On the bipanpositionable bipanconnectedness of hypercubes

A bipartite graph G is bipanconnected if, for any two distinct vertices x and y of G, it contains an [x, y]-path of length l for each integer l satisfying dG(x, y) ≤ l ≤ |V (G)| − 1 and 2|(l − dG(x, y)), where dG(x, y) denotes the distance between vertices x and y in G and V (G) denotes the vertex set of G. We say a bipartite graph G is bipanpositionably bipanconnected if, for any two distinct vertices x and y of G and for any vertex z ∈ V (G) − {x, y}, it contains a path Pl,k of length l such that x is the beginning vertex of Pl,k, z is the (k+ 1)-th vertex of Pl,k, and y is the ending vertex of Pl,k for each integer l satisfying dG(x, z)+ dG(y, z) ≤ l ≤ |V (G)| − 1 and 2|(l− dG(x, z)− dG(y, z)) and for each integer k satisfying dG(x, z) ≤ k ≤ l−dG(y, z) and 2|(k−dG(x, z)). In this paper, we prove that an ncube is bipanpositionably bipanconnected if n ≥ 4. As a consequence, many properties of hypercubes, such as bipancyclicity, bipanconnectedness, bipanpositionable Hamiltonicity, etc., follow directly from our result. © 2008 Elsevier B.V. All rights reserved.