Degree conditions and path factors with inclusion or exclusion properties

A spanning subgraph F of a graph G is called a path factor if every component of F is a path. For an integer d = 2, a P=d-factor of a graph G is a spanning subgraph F such that every component is isomorphic to a path of k vertices for some k = d. A graph G is called a P=d-factor covered graph if for any e ? E(G), G has a P=d-factor covering e. A graph G is called a P=d-factor deleted graph if for any e ? E(G), G has a P=d-factor excluding e. In this article, we verify that (i) a k-connected graph G with at least n vertices admits a P=3-factor if G satisfies max{dG(x(1)), dG(x(2)), . . ., dG(x(2k+1))} = n3 for any independent subset {x(1), x(2), . . ., x(2k+1)} of G, where k = 1 and n = 4k + 4 are two integers; (ii) a k-connected graph G with at least n vertices is a P=3-factor covered graph if G satisfies max{dG(x(1)), dG(x(2)), . . ., dG(x(2k-1))} = n+23 for any independent subset {x(1), x(2), . . ., x(2k-1)} of G, where k = 1 and n = 4k + 2 are two integers; (iii) a (k + 1)-connected graph G with at least n vertices is a P=3-factor deleted graph if G satisfies max{dG(x(1)), dG(x2), . . ., dG(x(2k-1))} = n3 for any independent subset {x(1), x(2), . . ., x(2k-1)} of G, where k = 1 and n = 4k + 2 are two integers.