The existence of path-factor uniform graphs with large connectivity

Abstract

A path-factor is a spanning subgraph F of G such that every component of F is a path with at least two vertices. Let k ≥ 2 be an integer. A P≥k-factor of G means a path factor in which each component is a path with at least k vertices. A graph G is a P≥k-factor covered graph if for any 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 e1, e2 ∈ E(G) with e1 ≠ e2, G has a P≥k-factor covering e1 and avoiding e2. In other words, a graph G is called a P≥k-factor uniform graph if for any e ∈ E(G), G − e is a P≥k-factor covered graph. In this paper, we present two sufficient conditions for graphs to be P≥3-factor uniform graphs depending on binding number and degree conditions. Furthermore, we show that two results are best possible in some sense.