Path Factors and Neighborhoods of Independent Sets in Graphs

A path-factor is a spanning subgraph F of G such that every component of F is a path with at least two vertices. Let k = 2 be an integer. A P-=k-factor of G means a path factor in which each component is a path with at least k vertices. A graph G is a P-=k-factor covered graph if for any e E E(G), G has a P-=k-factor including e. Let 0 be a real number with (1)/(3) =0 = 1 and k be a positive integer. We verify that (i) a k -connected graph G of order n with n > 5k + 2 has a P-=3-factor if |N-G(I)| > 0(n- 3k - 1) + k for every independent set I of G with |I| = [0(2k + 1)]; (ii) a (k + 1)-connected graph G of order n with n = 5k +2 is a P(=3-)factor covered graph if |N-G(I)| > 0(n - 3k - 1) + k + 1 for every independent set I of G with |I| = [0(2k + 1)].