  A graph is said to be $K_{1,r}$-free if it does not contain an induced
subgraph isomorphic to $K_{1,r}$. An $\mathcal{F}$-factor is a spanning
subgraph $H$ such that each connected component of $H$ is isomorphic to some
graph in $\mathcal{F}$. In particular, $H$ is called an $\{P_2,P_3\}$-factor of
$G$ if $\mathcal{F}=\{P_2,P_3\}$; $H$ is called an $\mathcal{S}_n$-factor of
$G$ if $\mathcal{F}=\{K_{1,1},K_{1,2},K_{1,3},...,K_{1,n}\}$, where $n\geq2$. A
spanning subgraph of a graph $G$ is called a $\mathcal{P}_{\geq k}$-factor of
$G$ if its each component is isomorphic to a path of order at least $k$, where
$k\geq2$. A graph $G$ is called a $\mathcal{F}$-factor covered graph if there
is a $\mathcal{F}$-factor of $G$ including $e$ for any $e\in E(G)$. In this
paper, we give a minimum degree condition for a $K_{1,r}$-free graph to have an
$\mathcal{S}_n$-factor and 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.
