Spectral radius and $[a,b]$-factors in graphs

TL;DR

This paper establishes spectral conditions for the existence of specific $[a,b]$-factors in graphs, extending previous results on perfect matchings and confirming a conjecture for certain graph sizes.

Contribution

It provides new spectral criteria for odd $[1,b]$-factors and $[a,b]$-factors, generalizing and improving prior theorems, and confirms a conjecture for large enough graphs.

Findings

Spectral conditions for odd $[1,b]$-factors in graphs.

Spectral conditions for $[a,b]$-factors in graphs.

Confirmation of a conjecture for graphs with size $n \\geq 3a+b-1$.

Abstract

An $[a,b]$-factor of a graph $G$ is a spanning subgraph $H$ such that $a\leq d_{H}(v)\leq b$ for each $v\in V(G)$. In this paper, we provide spectral conditions for the existence of an odd $[1,b]$-factor in a connected graph with minimum degree $\delta$ and the existence of an $[a,b]$-factor in a graph, respectively. Our results generalize and improve some previous results on perfect matchings of graphs. For $a=1$, we extend the result of O\cite{S.O} to obtain an odd $[1,b]$-factor and further improve the result of Liu, Liu and Feng\cite{W.L} for $a=b=1$. For $n\geq 3a+b-1$, we confirm the conjecture of Cho, Hyun, O and Park\cite{E.C}. We conclude some open problems in the end.