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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.

Some degree conditions for 𝒫≥k-factor covered graphs

Some degree conditions for 𝒫_≥k-factor covered graphs