The bipanpositionable bipancyclic property of the hypercubeI,II

A bipartite graph is bipancyclic if it contains a cycle of every even length from 4 to jV.G/j inclusive. A hamiltonian bipartite graph G is bipanpositionable if, for any two different vertices x and y, there exists a hamiltonian cycle C of G such that dC .x; y/ D k for any integer k with dG.x; y/ k jV.G/j=2 and .k􀀀dG.x; y// being even. A bipartite graph G is k-cycle bipanpositionable if, for any two different vertices x and y, there exists a cycle of G with dC .x; y/ D l and jV.C/j D k for any integer l with dG.x; y/ l k2and .l􀀀dG.x; y// being even. A bipartite graph G is bipanpositionable bipancyclic if G is k-cycle bipanpositionable for every even integer k, 4 k jV.G/j. We prove that the hypercube Qn is bipanpositionable bipancyclic for n 2.