Some degree conditions for 𝒫≥k-factor covered graphs

Guowei Dai,Yicheng Hang,Zan-Bo Zhang,Xiaoyan Zhang

doi:10.1051/ro/2021140

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.

Highlights

The graphs considered here are finite and simple, unless explicitly stated
A spanning subgraph of G is a subgraph H of G such that V (H) = V (G) and E(H) ⊆ E(G)
A subgraph H of G is called an induced subgraph of G if every pair of vertices in H which are adjacent in G are adjacent in H

Summary

Introduction

The graphs considered here are finite and simple, unless explicitly stated. Let G = (V (G), E(G)) be a graph. We use ω(G), i(G) to denote the number of components and isolated vertices of a graph G, respectively. School of Mathematical Science & Institute of Mathematics, Nanjing Normal University, Nanjing, Jiangsu210023, P.R. China. School of Statistics & Mathematics, and Institute of Artificial Intelligence & Deep Learning, Guangdong University of Finance & Economics, Guangzhou, Guangdong510630, P.R. China.

Results

Conclusion