Geodesic pancyclicity of crossed cubes

For a pair of vertices u, v ∈ V (G), a cycle is called a geodesic cycle with u and v if a shortest path of G joining u and v lies on the cycle. A graph G is pancyclic [12] if it contains a cycle of every length from 3 to |V (G)| inclusive. Furthermore, a graph G is called geodesic k-pancyclic [3] if for each pair of vertices u, v ∈ V (G), it contains a geodesic cycle of every integer length of l satisfying 2dG(u, v) + k ≤ l ≤ |V (G)|. Chang et al. [4] proved that CQn is pancyclic in the sense that a cycle of length l exists, 4 ≤ l ≤ |V (CQn)|. In this paper, we study a new pancyclic property and show that Crossed cubes is geodesic 4-pancyclic. Key–Words:crossed cubes, panconnected, pancyclic, geodesic cycle, geodesic pancyclic.