Isolated toughness and path-factor uniform graphs (II)

spanning subgraph F of G is called a path-factor if each component of F is a path. A P_≥ k -factor of G means a path-factor such that each component is a path with at least k vertices, where k≥ 2 is an integer. A graph G is called a P_≥ k -factor covered graph if for each e∈ E(G) , G has a P_≥ k -factor covering e . A graph G is called a P_≥ k -factor uniform graph if for any two different edges e_1,e_2∈ E(G) , G has a P_≥ k -factor covering e_1 and avoiding e_2 . In other word, a graph G is called a P_≥ k -factor uniform graph if for any e∈ E(G) , the graph G-e is a P_≥ k -factor covered graph. In this article, we demonstrate that (i) an (r+3) -edge-connected graph G is a P_≥ 2 -factor uniform graph if its isolated toughness I(G)>r+3/2r+3 , where r is a nonnegative integer; (ii) an (r+3) -edge-connected graph G is a P_≥ 3 -factor uniform graph if its isolated toughness I(G)>3r+6/2r+3 , 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.