Some results about star-factors in graphs

For a set $\mathcal{S}$ of connected graphs, a subgraph $F$ of a graph $G$ is defined as an $\mathcal{S}$-factor of $G$ if $F$ satisfies that $V(F)=V(G)$ and every component of $F$ is isomorphic to an element of $\mathcal{S}$. If every component of $F$ is a star, then $F$ is said to be a star-factor. A star-factor with size at most $n$ may be written for a $\{K_{1,t}: 1\leq t\leq n\}$-factor. A graph $G$ is called a $\{K_{1,t}: 1\leq t\leq n\}$-factor deleted graph if $G-e$ has a $\{K_{1,t}: 1\leq t\leq n\}$-factor for every $e\in E(G)$. The sun toughness of a graph $G$ is denoted by $s(G)$ and defined as follows :$$ s(G)=\min \big\{\frac{|X|}{sun(G-X)}: X\subseteq V(G), \ sun(G-X)\geq2 \big\} $$ if $G$ is not a complete graph, and $s(G)=+\infty$ if $G$ is a complete graph, where $sun(G-X)$ denotes the number of sun components of $G-X$. In this paper, we prove that (1) if $G$ is a connected graph, and its sun toughness satisfies $s(G)\geq\frac{1}{n}$, then $G$ admits a $\{K_{1,t}: 1\leq t\leq n\}$-factor; (2) if $G$ is a $(k+1)$-connected graph, and its sun toughness $s(G)>\frac{k+1}{n+1}$, then $G-Y$ admits a $\{K_{1,t}: 1\leq t\leq n\}$-factor for any $Y\subseteq V(G)$ with $|Y|=k$; (3) if $G$ is a 2-edge-connected graph, and its sun toughness $s(G)\geq\frac{1}{n-1}$, then $G$ is a $\{K_{1,t}: 1\leq t\leq n\}$-factor deleted graph. Furthermore, it is shown that our results are sharp.