Isolated toughness for path factors in networks

¹ School of Public Management, Jiangsu University of Science and Technology, Zhenjiang, Jiangsu 212100, P.R. China
² School of Economics and Management, Wenzhou University of Technology, Wenzhou, Zhejiang 325000, P.R. China

^* Corresponding author: wangsufangjust@163.com

Received: 23 December 2021
Accepted: 18 July 2022

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.