Remarks on component factors in K_1,r-free graphs

¹ College of Science, Nanjing Forestry University, Nanjing, Jiangsu 210037, P.R. China
² School of Statistics & Mathematics, and Institute of Artificial Intelligence & Deep Learning, Guangdong University of Finance & Economics, Guangzhou, Guangdong 510630, P.R. China
³ School of Mathematical Science & Institute of Mathematics, Nanjing Normal University, Nanjing, Jiangsu 210023, P.R. China

^* Corresponding author: daiguowei@njfu.edu.cn

Received: 7 October 2022
Accepted: 29 March 2023

Abstract

An ℱ-factor is a spanning subgraph H such that each connected component of H is isomorphic to some graph in ℱ. We use P_k and K_1,r to denote the path of order k and the star of order r + 1, respectively. In particular, H is called a {P₂, P₃}-factor of G if ℱ = {P₂, P₃}; H is called a P≥_k-factor of G if ℱ = {P_k, P_k+1,…}, where k ≥ 2; H is called an S_n-factor of G if ℱ = {P₂, P₃, K_1,3,…, K_1,n}, where n ≥ 2. A graph G is called a ℱ≥_k-factor covered graph if there is a ℱ≥_k-factor of G including e for any e ∈ E(G). We call a graph G is K_1,r-free if G does not contain an induced subgraph isomorphic to K_1,r. In this paper, we give a minimum degree condition for the K_1,r-free graph with an S_n-factor and the K_1,r-free graph with a ℱ≥₃-factor, respectively. Further, we obtain sufficient conditions for K_1,r-free graphs to be ℱ≥₂-factor, ℱ≥₃-factor or {P₂, P₃}-factor covered graphs. In addition, examples show that our results are sharp.