The existence of path-factor uniform graphs with large connectivity

Abstract

A path-factor is a spanning subgraphFofGsuch that every component ofFis a path with at least two vertices. Letk ≥ 2 be an integer. AP≥k-factor ofGmeans a path factor in which each component is a path with at leastkvertices. A graphGis aP≥k-factor covered graph if for anye ∈ E(G),Ghas aP≥k-factor coveringe. A graphGis called aP≥k-factor uniform graph if for anye1, e2 ∈ E(G) withe1 ≠ e2,Ghas aP≥k-factor coveringe1and avoidinge2. In other words, a graphGis called aP≥k-factor uniform graph if for anye ∈ E(G),G − eis aP≥k-factor covered graph. In this paper, we present two sufficient conditions for graphs to beP≥3-factor uniform graphs depending on binding number and degree conditions. Furthermore, we show that two results are best possible in some sense.