Let 𝒜 be a set of connected graphs. Then a spanning subgraph A of G is called an 𝒜-factor if each component of A is isomorphic to some member of 𝒜. Especially, when every graph in 𝒜 is a path, A is a path factor. For a positive integer d ≥ 2, we write 𝒫 ≥ d = {𝒫 i | i ≥ d}. Then a 𝒫 ≥ d -factor means a path factor in which every component admits at least d vertices. A graph G is called a (𝒫 ≥ d , m)-factor deleted graph if G – E′ admits a 𝒫 ≥ d -factor for any E′ ⊆ E( G) with | E′| = m. A graph G is called a (𝒫 ≥ d , k)-factor critical graph if G – Q has a 𝒫 ≥ d -factor for any Q ⊆ V ( G) with | Q| = k. In this paper, we present two degree conditions for graphs to be (𝒫 ≥3 , m)-factor deleted graphs and (𝒫 ≥3 , k)-factor critical graphs. Furthermore, we show that the two results are best possible in some sense.

In communication networks, the binding numbers of graphs (or networks) are often used to measure the vulnerability and robustness of graphs (or networks). Furthermore, the fractional factors of graphs and the fractional ID-[a, b]-factor-critical covered graphs have a great deal of important applications in the data transmission networks. In this paper, we investigate the relationship between the binding numbers of graphs and the fractional ID-[a, b]-factor-critical covered graphs, and derive a binding number condition for a graph to be fractional ID-[a, b]-factor-critical covered, which is an extension of Zhou’s previous result [S. Zhou, Binding numbers for fractional ID-k-factor-critical graphs, Acta Mathematica Sinica, English Series 30(1)(2014)181–186].