Some degree conditions for

A spanning subgraph of a graph G is called a path-factor of G if its each component is a path. A path-factor is called a ≥k-factor of G if its each component admits at least k vertices, where k ≥ 2. (Zhang and Zhou, Discrete Math. 309 (2009) 2067–2076) defined the concept of ≥k-factor covered graphs, i.e., G is called a ≥k-factor covered graph if it has a ≥k-factor covering e for any e∈ E(G). In this paper, we firstly obtain a minimum degree condition for a planar graph being a ≥2-factor and ≥3-factor covered graph, respectively. Secondly, we investigate the relationship between the maximum degree of any pairs of non-adjacent vertices and ≥k-factor covered graphs, and obtain a sufficient condition for the existence of ≥2-factor and ≥3-factor covered graphs, respectively