Remarks on component factors in K1,r-free graphs

An $\mathcal{F}$-factor is a spanning subgraph $H$ such that each connected component of $H$ is isomorphic to some graph in $\mathcal{F}$. We use $P_k$ and $K_{1,r}$ to denote the path of order $k$ and the star of order $r+1$, respectively. In particular, $H$ is called a $\{P_2,P_3\}$-factor of $G$ if $\mathcal{F}=\{P_2,P_3\}$; $H$ is called a $\mathcal{P}_{\geq k}$-factor of $G$ if $\mathcal{F}=\{P_2,P_3,...,P_k\}$, where $k\geq2$; $H$ is called an $\mathcal{S}_n$-factor of $G$ if $\mathcal{F}=\{P_2,P_3,K_{1,3},...,K_{1,n}\}$, where $n\geq2$. A graph $G$ is called a $\mathcal{P}_{\geq k}$-factor covered graph if there is a $\mathcal{P}_{\geq k}$-factor of $G$ including $e$ for any $e\in E(G)$. We call a graph $G$ is $K_{1,r}$-free if $G$ does not contain an induced subgraph isomorphic to $K_{1,r}$. In this paper, we give a minimum degree condition for the $K_{1,r}$-free graph with an $\mathcal{S}_n$-factor and the $K_{1,r}$-free graph with a $\mathcal{P}_{\geq 3}$-factor, respectively. Further, we obtain sufficient conditions for $K_{1,r}$-free graphs to be $\mathcal{P}_{\geq 2}$-factor, $\mathcal{P}_{\geq 3}$-factor or $\{P_2,P_3\}$-factor covered graphs. In addition, examples show that our results are sharp.