Some degree conditions for 𝒫_≥k-factor covered graphs

¹ School of Mathematical Science & Institute of Mathematics, Nanjing Normal University, Nanjing, Jiangsu 210023, 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

^* Corresponding author: zhangxiaoyan@njnu.edu.cn; zhang_njnu@aliyun.com

Received: 15 December 2020
Accepted: 15 September 2021

Abstract

A spanning subgraph of a graph G is called a path-factor of G if its each component is a path. A path-factor is called a 𝒫_≥k-factor of G if its each component admits at least k vertices, where k ≥ 2. (Zhang and Zhou, Discrete Math. 309 (2009) 2067–2076) defined the concept of 𝒫_≥k-factor covered graphs, i.e., G is called a 𝒫_≥k-factor covered graph if it has a 𝒫_≥k-factor covering e for any e∈ E(G). In this paper, we firstly obtain a minimum degree condition for a planar graph being a 𝒫_≥2-factor and 𝒫_≥3-factor covered graph, respectively. Secondly, we investigate the relationship between the maximum degree of any pairs of non-adjacent vertices and 𝒫_≥k-factor covered graphs, and obtain a sufficient condition for the existence of 𝒫_≥2-factor and 𝒫_≥3-factor covered graphs, respectively.