Isolated toughness for path factors in networks

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.