Sun toughness and path-factor uniform graphs

Abstract

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 e1 and e2 of G, G has a P≥k-factor covering e1 and excluding e2. 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)>4/5.