Volume 55, Number 5, September-October 2021
|Page(s)||2907 - 2913|
|Published online||08 October 2021|
Some degree conditions for 𝒫≥k-factor covered graphs
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
Accepted: 15 September 2021
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.
Mathematics Subject Classification: 05C70 / 05C38
Key words: Graph / 𝒫≥2-factor covered graph / 𝒫≥3-factor covered graph / planar graph / Minimum degree
© The authors. Published by EDP Sciences, ROADEF, SMAI 2021
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