Geodesic-Bipancyclicity of Hypercubes

A shortest path connecting two vertices u and v is called a u-v geodesic. The distance between u and v in a graph G, denoted by d_G(u, v), is the number of edges in a u-v geodesic. In this paper, a property called geodesic-bipancyclicity on bipartite graphs is introduced. A bipartite graph G with n vertices is geodesic-bipancyclic if, for each pair of vertices u, v є V(G) and for each even integer l satisfying max{2d_G(u,v),4} ≤ l ≤ n, every u-v geodesic lies on a cycle of length l. I start with an illustration about the relationship between geodesic-bipancyclicity and other Hamiltonian-like properties. Then, it shows that hypercubes are geodesic-bipancyclic.