Given a graph G and an integer \(k\ge 2\). A spanning subgraph F of a graph G is said to be a \(P_{\ge k}\)-factor of G if each component of F is a path of order at least k. A graph G is called a \(P_{\ge k}\)-factor uniform graph if for any two distinct edges \(e_{1}\) and \(e_{2}\) of G, G admits a \(P_{\ge k}\)-factor including \(e_{1}\) and excluding \(e_{2}\). More recently, Zhou and Sun (Discret Math 343:111715, 2020) gave binding number conditions for a graph to be \(P_{\ge 2}\)-factor and \(P_{\ge 3}\)-factor uniform graphs, respectively. In this paper, we present toughness and isolated toughness conditions for a graph to be a \(P_{\ge 3}\)-factor uniform graph, respectively.

给定一个图G和一个整数\（k \ ge 2 \）。甲生成子图˚F的曲线图的ģ被说成是一个\（P _ {\锗的K} \）的α-因子ģ如果的每个分量˚F是至少顺序的路径ķ。一个图ģ称为\（P _ {\锗的K} \） -因子均匀图表，如果对任意两个不同的边缘\（E_ {1} \）和\（E_ {2} \）的G ^，ģ承认一个\ （P _ {\ ge k} \）-包括\（e_ {1} \）且不包括\（e_ {2} \）的因数。最近，Zhou和Sun（Discret Math 343：111715，2020）给出了图的绑定数条件为\（P _ {\ ge 2} \）-因子和\（P _ {\ ge 3} \）-因子均匀的条件。图。在本文中，我们给出了图分别为\（P _ {\ ge 3} \）因数均匀图的韧性和孤立韧性条件。