Geodesic and Balanced Bipancyclicity of Hypercubes

For any two vertices u, v ∈ V (G), a cycle C in G is called a geodesic cycle between u and v if a shortest path of G joining u and v lies on the cycle. Let G be a bipartite graph. For any two vertices u and v in G, a cycle C is called a balanced cycle between u and v if dC(u, v) = max{dC(x, y) | x and u are in the same partite set, and y and v are in the same partite set }. A bipartite graph G is geodesic bipancyclic (respectively, balanced bipancyclic) if for each pair of vertices u, v ∈ V (G), it contains a geodesic cycle (respectively, balanced cycle) of every even length of k satisfying max{2dG(u, v), 4} ≤ k ≤ |V (G)| between u and v. In this paper, we prove that Qn is geodesic bipancyclic and balanced bipancyclic if n ≥ 2. Key–Words: Hypercube; Interconnection networks; Edge-bipancyclic; Geodesic bipancyclic; Balanced bipancyclic