For a graph G and an integer \(t \ge 2\) , a spanning subgraph H of G is called a \(P_{\ge t}\) -factor of G if each component of H is a path of order at least t. If for any edge \(e \in E(G)\) , there is a \(P_{\ge t}\) -factor of G containing e, then we call G a \(P_{\ge t}\) -factor covered graph. Furthermore, if for any two distinct edges \(e_1\) and \(e_2\) of E(G), there is a \(P_{\ge t}\) -factor of G containing \(e_1\) and excluding \(e_2\) , then we call G a \(P_{\ge t}\) -factor uniform graph. In this note, we establish some sufficient conditions related to toughness and isolated toughness for graphs (under degree constraints) to be \(P_{\ge 3}\) -factor covered or \(P_{\ge 3}\) -factor uniform. We also show the conditions are best possible. Our results improve some known results in the literature.
