Regular connected bipancyclic spanning subgraphs of hypercubes

An nn-dimensional hypercube QnQn is a Hamiltonian graph; in other words QnQn (n≥2n≥2) contains a spanning subgraph which is 2-regular and 2-connected. In this paper, we explore yet another strong property of hypercubes. We prove that for any integer kk with 3≤k≤n3≤k≤n, QnQn (n≥3n≥3) contains a spanning subgraph which is kk-regular, kk-connected and bipancyclic. We also obtain the result that every mesh Pm×PnPm×Pn (m,n≥2m,n≥2) is bipancyclic, which is used to prove the property above.