Toughness and isolated toughness conditions for $P_{\ge 3}$-factor uniform graphs

Abstract

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.