Some Existence Theorems on Path Factors with Given Properties in Graphs - Zhou, Si Zhong; Sun, Zhi Ren (Acta mathematica Sinica. English series, 2020)
A path factor of G is a spanning subgraph of G such that its each component is a path. A path factor is called a P ≥ n -factor if its each component admits at least n vertices. A graph G is called P ≥ n -factor covered if G admits a P ≥ n -factor containing e for any e ∈ E ( G ), which is defined by [ Discrete Mathematics , 309 , 2067–2076 (2009)]. We first define the concept of a ( P ≥ n , k )-factor-critical covered graph, namely, a graph G is called ( P ≥ n , k )-factor-critical covered if G-D is P ≥ n -factor covered for any D ⊆ V ( G )with ∣ D ∣ = k . In this paper, we verify that (i) a graph G with k ( G ) ≥ k + 1 is ( P ⊆ 2 , k )-factor-critical covered if bind ( G ) > 2 + k 3 ; (ii) a graph G with ∣ V ( G )∣ ≥ k + 3 and k ( G ) ≥ k + 1 is ( P ≥ 3 , k )-factor-critical covered if bind ( G ) ≥ 4 + k 3 .