Toughness and isolated toughness conditions for path-factor critical covered graphs

Abstract

Given a graph $G$ and an integer $k\geq2$. A spanning subgraph $H$ of $G$ is called a $P_{\geq k}$-factor of $G$ if every component of $H$ is a path with at least $k$ vertices. A graph $G$ is said to be $P_{\geq k}$-factor covered if for any $e\in E(G)$, $G$ admits a $P_{\geq k}$-factor including $e$. A graph $G$ is called a $(P_{\geq k},n)$-factor critical covered graph if $G-V'$ is $P_{\geq k}$-factor covered for any $V'\subseteq V(G)$ with $|V'|=n$. In this paper, we study the toughness and isolated toughness conditions for $(P_{\geq k},n)$-factor critical covered graphs, where $k=2,3$. Let $G$ be a $(n+1)$-connected graph. It is shown that (i) $G$ is a $(P_{\geq 2},n)$-factor critical covered graph if its toughness $\tau(G)>\frac{n+2}{3}$; (ii) $G$ is a $(P_{\geq 2},n)$-factor critical covered graph if its isolated toughness $I(G)>\frac{n+1}{2}$; (iii) $G$ is a $(P_{\geq 3},n)$-factor critical covered graph if $\tau(G)>\frac{n+2}{3}$ and $|V(G)|\geq n+3$; (iv) $G$ is a $(P_{\geq 3},n)$-factor critical covered graph if $I(G)>\frac{n+3}{2}$ and $|V(G)|\geq n+3$.  Furthermore, we claim that these conditions are best possible in some sense.