Remarks on component factors in K1,r-free graphs

Abstract

An ℱ-factor is a spanning subgraph H such that each connected component of H is isomorphic to some graph in ℱ. We use Pk and K1,r to denote the path of order k and the star of order r + 1, respectively. In particular, H is called a {P2, P3}-factor of G if ℱ = {P2, P3}; H is called a P≥k-factor of G if ℱ = {Pk, Pk+1,…}, where k ≥ 2; H is called an Sn-factor of G if ℱ = {P2, P3, K1,3,…, K1,n}, where n ≥ 2. A graph G is called a ℱ≥k-factor covered graph if there is a ℱ≥k-factor of G including e for any e ∈ E(G). We call a graph G is K1,r-free if G does not contain an induced subgraph isomorphic to K1,r. In this paper, we give a minimum degree condition for the K1,r-free graph with an Sn-factor and the K1,r-free graph with a ℱ≥3-factor, respectively. Further, we obtain sufficient conditions for K1,r-free graphs to be ℱ≥2-factor, ℱ≥3-factor or {P2, P3}-factor covered graphs. In addition, examples show that our results are sharp.