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.