A spanning subgraph
![$F$](img1.png)
of a graph
![$G$](img2.png)
is called a path factor if every component of
![$F$](img1.png)
is a path. For an integer
![$d\geq2$](img3.png)
, a
![$P_{\geq d}$](img4.png)
-factor of a graph
![$G$](img2.png)
is a spanning subgraph
![$F$](img1.png)
such that every component is isomorphic to a path of
![$k$](img5.png)
vertices for some
![$k\geq d$](img6.png)
. A graph
![$G$](img2.png)
is called a
![$P_{\geq d}$](img4.png)
-factor covered graph if for any
![$e\in E(G)$](img7.png)
,
![$G$](img2.png)
has a
![$P_{\geq d}$](img4.png)
-factor covering
![$e$](img8.png)
. A
graph
![$G$](img2.png)
is called a
![$P_{\geq d}$](img4.png)
-factor deleted graph if for any
![$e\in E(G)$](img7.png)
,
![$G$](img2.png)
has a
![$P_{\geq d}$](img4.png)
-factor excluding
![$e$](img8.png)
. In this article,
we verify that (i) a
![$k$](img5.png)
-connected graph
![$G$](img2.png)
with at least
![$n$](img9.png)
vertices admits a
![$P_{\geq3}$](img10.png)
-factor if
![$G$](img2.png)
satisfies
![$\max\{d_G(x_1),d_G(x_2),\cdots,d_G(x_{2k+1})\}\geq\frac{n}{3}$](img11.png)
for any independent subset
![$\{x_1,x_2,\cdots,x_{2k+1}\}$](img12.png)
of
![$G$](img2.png)
, where
![$k\geq1$](img13.png)
and
![$n\geq4k+4$](img14.png)
are two integers; (ii) a
![$k$](img5.png)
-connected graph
![$G$](img2.png)
with at least
![$n$](img9.png)
vertices is a
![$P_{\geq3}$](img10.png)
-factor
covered graph if
![$G$](img2.png)
satisfies
![$\max\{d_G(x_1),d_G(x_2),\cdots,d_G(x_{2k-1})\}\geq\frac{n+2}{3}$](img15.png)
for any independent subset
![$\{x_1,x_2,\cdots,x_{2k-1}\}$](img16.png)
of
![$G$](img2.png)
, where
![$k\geq1$](img13.png)
and
![$n\geq4k+2$](img17.png)
are two integers; (iii) a
![$(k+1)$](img18.png)
-connected graph
![$G$](img2.png)
with
at least
![$n$](img9.png)
vertices is a
![$P_{\geq3}$](img10.png)
-factor deleted graph if
![$G$](img2.png)
satisfies
![$\max\{d_G(x_1),d_G(x_2),\cdots,d_G(x_{2k-1})\}\geq\frac{n}{3}$](img19.png)
for any independent subset
![$\{x_1,x_2,\cdots,x_{2k-1}\}$](img16.png)
of
![$G$](img2.png)
, where
![$k\geq1$](img13.png)
and
![$n\geq4k+2$](img17.png)
are two integers.