Sun toughness and path-factor uniform graphs

Abstract

A path-factor is a spanning subgraphFofGsuch that each component ofFis a path of order at least two. Letkbe an integer withk ≥ 2. AP≥k-factor is a spanning subgraph ofGwhose components are paths of order at leastk. A graphGis called aP≥k-factor covered graph if for any edgeeofG,Gadmits aP≥k-factor coveringe. A graphGis called aP≥k-factor uniform graph if for any two distinct edgese1ande2ofG,Ghas aP≥k-factor coveringe1and excludinge2. In this article, we claim that (1) a 4-edge-connected graphGis aP≥3-factor uniform graph if its sun toughnesss(G) ≥ 1; (2) a 4-connected graphGis aP≥3-factor uniform graph if its sun toughnesss(G)>4/5.