Remarks on component factors in graphs

School of Mathematical Sciences, Nanjing Normal University, Nanjing, Jiangsu 210023, P.R. China

^* Corresponding author: guowei_dai@aliyun.com

Received: 16 September 2021
Accepted: 27 February 2022

Abstract

For a family of connected graphs ℱ, a spanning subgraph H of a graph G is called an ℱ-factor of G if its each component is isomorphic to an element of ℱ. In particular, H is called an S_k-factor of G if ℱ = {K_1,1, K_1,2,…,K_1,k}, where integer k ≥ 2; H is called a P_≥3-factor of G if every component in ℱ is a path of order at least three. As an extension of S_k-factors, the induced star-factor (i.e., IS_k-factor) is a spanning subgraph each component of which is an induced subgraph isomorphic to some graph in ℱ = {K_1,1, K_1,2,…,K_1,k}. In this paper, we firstly prove that a graph G has an S_k-factor if and only if its isolated toughness I(G) ≥ 1/k. Secondly, we prove that a planar graphs G has an S₂-factors if its minimum degree δ(G) ≥ 3. Thirdly, we give two sufficient conditions for graphs with IS_k-factors by toughness and minimum degree, respectively. Additionally, we obtain three special classes of graphs admitting P_≥3-factors.