Isolated toughness and path-factor uniform graphs (II)

A 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 2 r + 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 ) > 3 r + 6 2 r + 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.