Weak-vertex-pancyclicity of (n, k)-star graphs

The (n,k)-star graph (Sn,k for short) is an attractive alternative to the hypercube and also a generalized version of the n-star. It is isomorphic to the n-star (n-complete) graph if k=n−1 (k=1). Jwo et al. have already demonstrated in 1991 that an n-star contains a cycle of every even length from 6 to n!. This work shows that every vertex in an Sn,k lies on a cycle of length l for every 3≤l≤n!/(n−k)! when 1≤k≤n−4 and n≥6. Additionally, for n−3≤k≤n−2, each vertex in an Sn,k is contained in a cycle of length ranged from 6 to n!/(n−k)!. Moreover, each constructed cycle of an available length in an Sn,k can contain a desired 1-edge.