Bipanpositionable Bipancyclic of Hypercube

Abstract

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 di®erent vertices x and y, there exists a hamiltonian cycle C of G such that dC(x; y) = 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 di®erent vertices x and y, there exists a cycle of G with dC(x; y) = l and jV (C)j = k and for any integer l with dG(x; y)≦l ≦ k/ 2 and (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 if and only if n ≥ 2.