Geodesic bipancyclicity of the Cartesian product of graphs

Abstract

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 $2d + 2\leq l \leq |V(G)|,$ 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 $n \geq 2$ every $n$-dimensional torus is a geodesic 2-bipancyclic graph.