A spanning subgraph F of G is called a path-factor if each component of F is a path. A P≥k\documentclass[12pt]{minimal}
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\begin{document}$$P_{\ge k}$$\end{document}-factor of G means a path-factor such that each component is a path with at least k vertices, where k≥2\documentclass[12pt]{minimal}
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\begin{document}$$k\ge 2$$\end{document} is an integer. A graph G is called a P≥k\documentclass[12pt]{minimal}
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\begin{document}$$P_{\ge k}$$\end{document}-factor covered graph if for each e∈E(G)\documentclass[12pt]{minimal}
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\begin{document}$$e\in E(G)$$\end{document}, G has a P≥k\documentclass[12pt]{minimal}
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\begin{document}$$P_{\ge k}$$\end{document}-factor covering e. A graph G is called a P≥k\documentclass[12pt]{minimal}
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\begin{document}$$P_{\ge k}$$\end{document}-factor uniform graph if for any two different edges e1,e2∈E(G)\documentclass[12pt]{minimal}
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\begin{document}$$e_1,e_2\in E(G)$$\end{document}, G has a P≥k\documentclass[12pt]{minimal}
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\begin{document}$$P_{\ge k}$$\end{document}-factor covering e1\documentclass[12pt]{minimal}
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\begin{document}$$e_1$$\end{document} and avoiding e2\documentclass[12pt]{minimal}
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\begin{document}$$e_2$$\end{document}. In other word, a graph G is called a P≥k\documentclass[12pt]{minimal}
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\begin{document}$$P_{\ge k}$$\end{document}-factor uniform graph if for any e∈E(G)\documentclass[12pt]{minimal}
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\begin{document}$$e\in E(G)$$\end{document}, the graph G-e\documentclass[12pt]{minimal}
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\begin{document}$$G-e$$\end{document} is a P≥k\documentclass[12pt]{minimal}
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\begin{document}$$P_{\ge k}$$\end{document}-factor covered graph. In this article, we demonstrate that (i) an (r+3)\documentclass[12pt]{minimal}
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\begin{document}$$(r+3)$$\end{document}-edge-connected graph G is a P≥2\documentclass[12pt]{minimal}
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\begin{document}$$P_{\ge 2}$$\end{document}-factor uniform graph if its isolated toughness I(G)>r+32r+3\documentclass[12pt]{minimal}
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\begin{document}$$I(G)>\frac{r+3}{2r+3}$$\end{document}, where r is a nonnegative integer; (ii) an (r+3)\documentclass[12pt]{minimal}
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\begin{document}$$(r+3)$$\end{document}-edge-connected graph G is a P≥3\documentclass[12pt]{minimal}
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\begin{document}$$P_{\ge 3}$$\end{document}-factor uniform graph if its isolated toughness I(G)>3r+62r+3\documentclass[12pt]{minimal}
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\begin{document}$$I(G)>\frac{3r+6}{2r+3}$$\end{document}, 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.
