A graph G has a P⩾k\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {P}}_{\geqslant k}$$\end{document}-factor if G has a spanning subgraph H such that every component of H is a path of order at least k. A graph G is P⩾k\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {P}}_{\geqslant k}$$\end{document}-factor deleted if G-e\documentclass[12pt]{minimal}
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\begin{document}$$G-e$$\end{document} has a P⩾k\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {P}}_{\geqslant k}$$\end{document}-factor for each edge e of G. In this paper, we give two necessary and sufficient conditions defining a P⩾2\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {P}}_{\geqslant 2}$$\end{document}-factor deleted graph and a P⩾3\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {P}}_{\geqslant 3}$$\end{document}-factor deleted graph, respectively. Based on the result of P⩾2\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {P}}_{\geqslant 2}$$\end{document}-factor deleted graphs, we establish respectively a lower bound on the size and a lower bound on the spectral radius to ensure whether or not a graph is P⩾2\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {P}}_{\geqslant 2}$$\end{document}-factor deleted. Furthermore, by constructing extremal graphs, we show that all the above bounds are best possible.
