Sun toughness and path-factor uniform graphs

A path-factor is a spanning subgraph F of G such that each component of F is a path of order at least two. Let k be an integer with k >= 2. A P->= k-factor is a spanning subgraph of G whose components are paths of order at least k. A graph G is called a P->= k-factor covered graph if for any edge e of G, G admits a P->= k-factor covering e. A graph G is called a P->= k-factor uniform graph if for any two distinct edges e(1) and e(2) of G, G has a P->= k-factor covering e(1) and excluding e(2). In this article, we claim that (1) a 4-edge-connected graph G is a P->= 3-factor uniform graph if its sun toughness s(G) >= 1; (2) a 4-connected graph G is a P->= 3-factor uniform graph if its sun toughness s(G)> 4/5s.