Isolated toughness and path-factor uniform graphs (II)

Abstract

A spanning subgraph F of G is called a path-factor if each component of F is a path. A $P_{\ge k}$-factor of G means a path-factor such that each component is a path with at least k vertices, where $k\ge 2$ is an integer. A graph G is called a $P_{\ge k}$-factor covered graph if for each $e\in E(G)$, G has a $P_{\ge k}$-factor covering e. A graph G is called a $P_{\ge k}$-factor uniform graph if for any two different edges $e_1,e_2\in E(G)$, G has a $P_{\ge k}$-factor covering $e_1$ and avoiding $e_2$. In other word, a graph G is called a $P_{\ge k}$-factor uniform graph if for any $e\in E(G)$, the graph $G-e$ is a $P_{\ge k}$-factor covered graph. In this article, we demonstrate that (i) an $(r+3)$-edge-connected graph G is a $P_{\ge 2}$-factor uniform graph if its isolated toughness $I(G)>\frac{r+3}{2r+3}$, where r is a nonnegative integer; (ii) an $(r+3)$-edge-connected graph G is a $P_{\ge 3}$-factor uniform graph if its isolated toughness $I(G)>\frac{3r+6}{2r+3}$, where r is a nonnegative integer. Furthermore, we claim that these conditions on isolated toughness and edge-connectivity in our main results are best possible in some sense.