The existence of path-factor covered graphs

A spanning subgraph H of a graph G is called a P≥k-factor of G if every component of H is isomorphic to a path of order at least k, where k ≥ 2. A graph G is called a P≥k-factor covered graph if there is a P≥k-factor of G covering e for any e ∈ E(G). In this paper, we obtain two special classes of P≥2-factor covered graphs. We also obtain two special classes of P≥3-factor covered graphs. Furthermore, it is shown that these results are all sharp.