Isolated toughness for path factors in networks
Sufang Wang, Wei Zhang
Abstract
Let ℋ be a set of connected graphs. Then an ℋ-factor is a spanning subgraph of G , whose every connected component is isomorphic to a member of the set ℋ. An ℋ-factor is called a path factor if every member of the set ℋ is a path. Let k ≥ 2 be an integer. By a P ≥ k -factor we mean a path factor in which each component path admits at least k vertices. A graph G is called a ( P ≥ k , n )-factor-critical covered graph if for any W ⊆ V ( G ) with | W | = n and any e ∈ E ( G − W ), G − W has a P ≥ k -factor covering e . In this article, we verify that (1) an ( n + λ + 2)-connected graph G is a ( P ≥2 , n )-factor-critical covered graph if its isolated toughness I ( G ) > n +λ+2/2λ+3, where n and λ are two nonnegative integers; (2) an ( n + λ + 2)-connected graph G is a ( P ≥3 , n )-factor-critical covered graph if its isolated toughness I ( G ) > n +3λ+5/2λ+3, where n and λ be two nonnegative integers.