Sufficient conditions for graphs with {P₂, P₅}-factors

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

^* Corresponding author: wenqiwangcc@gmail.com

Received: 6 September 2022
Accepted: 24 June 2022

Abstract

For a graph G, a spanning subgraph F of G is called an {P₂, P₅}-factor if every component of F is isomorphic to P₂ or P₅, where P_k denotes the path of order k. It was proved by Egawa and Furuya that if G satisfies 3c₁ (G − S) + 2c₃ (G − S) ≤ 4|S| + 1 for all S ⊆ V(G), then G has a {P₂, P₅}-factor, where c_k (G − S) denotes the number of components of G − S with order k. By this result, we give some other sufficient conditions for a graph to have a {P₂, P₅}-factor by various graphic parameters such as toughness, binding number, degree sums, etc. Moreover, we obtain some regular graphs and some K_(1,r)-free graphs having {P₂, P₅}-factors.