Geodesic bipancyclicity of the Cartesian product of graphs

A cycle containing a shortest path between two vertices u and v in a graph G is called a ( u, v )-geodesic cycle. A connected graph G is geodesic 2-bipancyclic, if every pair of vertices u, v of it is contained in a ( u, v )-geodesic cycle of length l for each even integer l satisfying 2 d + 2 ≤ l ≤ | V ( G ) | , where d is the distance between u and v. In this paper, we prove that the Cartesian product of two geodesic hamiltonian graphs is a geodesic 2-bipancyclic graph. As a consequence, we show that for n ≥ 2 every n -dimensional torus is a geodesic 2-bipancyclic graph.