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

Abstract

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 $P_{\geq n}$-factor
covered graph if $G$ admits a $P_{\geq n}$-factor containing $e$ for each $e\in E(G)$. In this
paper, 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 a
graph 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.