Two sufficient conditions for odd [1,b]-factors in graphs

• Published in: Linear algebra and its applications Vol. 661; pp. 149 - 162
• Main Authors: Zhou, Sizhong, Liu, Hongxia
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.03.2023
• Subjects: Graph; Odd [formula omitted]-factor; Size; Spectral radius; Graph; 05C50; Size; Spectral radius; Odd [1,b]-factor; 05C70; 05C35
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2022.12.018

An odd[1,b]-factor of a graph G is a spanning subgraph F of G with dF(x)∈{1,3,⋯,b} for every x∈V(G), where b is a positive odd integer. Let |E(G)| be the size of G, and let ρ(G) be the spectral radius of G. In this article, we first verify three lower bounds for |E(G)| in a graph G of even order n to guarantee the existence of an odd [1,b]-factor in G. Then we prove two lower bounds for ρ(G) in a graph G of even order n to guarantee the existence of an odd [1,b]-factor in G. Furthermore, we create some extremal graphs to show all the lower bounds derived in this article are sharp.