Sun toughness and $P_{\geq3}$-factors in graphs

A $P_{\geq n}$-factor means a path factor with each component having at least $n$ vertices,where $n\geq2$ is an integer. A graph $G$ is called a $P_{\geq n}$-factor deleted graph if $G-e$admits a $P_{\geq n}$-factor for any $e\in E(G)$. A graph $G$ is called a $P_{\geq n}$-factorcovered graph if $G$ admits a $P_{\geq n}$-factor containing $e$ for each $e\in E(G)$. In thispaper, we first introduce a new parameter, i.e., sun toughness, which is denoted by $s(G)$. $s(G)$is defined as follows:$$s(G)=\min\{\frac{|X|}{sun(G-X)}: X\subseteq V(G), \ sun(G-X)\geq2\}$$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 $G-X$. Then we obtain two sun toughness conditions for agraph to be a $P_{\geq n}$-factor deleted graph or a $P_{\geq n}$-factor covered graph. Furthermore,it is shown that our results are sharp.