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.
Funder
National Natural Science Foundation of China
Natural Science Foundation of Guangdong Province
Qinglan Project of Jiangsu Province of China
Subject
Management Science and Operations Research,Computer Science Applications,Theoretical Computer Science
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