Geodesic pancyclicity of twisted cubes

Abstract

The hypercube is one of the most popular interconnection networks since it has simple structure and is easy to implement. An n-dimensional twisted cube, TQ n , is an important variation of the hypercube Q n and preserves many of its desirable properties. The problem of embedding linear arrays and cycles into a host graph has attracted substantial attention in recent years. The geodesic cycle embedding problem is for any two distinct vertices, to find all the possible lengths of cycles that include a shortest path joining them. In this paper, we prove that TQ n is geodesic 2-pancyclic for each odd integer n ⩾ 3. This result implies that TQ n is edge-pancyclic for each odd integer n ⩾ 3. Moreover, TQ n × K 2 is also demonstrated to be geodesic 4-pancyclic.