Spectral conditions for graphs having all (fractional) $[a,b]$-factors

TL;DR

This paper establishes precise spectral radius conditions that guarantee graphs possess all $[a,b]$-factors and fractional $[a,b]$-factors, extending factor theory with spectral graph analysis.

Contribution

It provides the first tight spectral radius criteria for ensuring graphs have all $[a,b]$-factors and fractional $[a,b]$-factors, broadening factor existence conditions.

Findings

Derived tight spectral radius bounds for $[a,b]$-factors.

Derived tight spectral radius bounds for fractional $[a,b]$-factors.

Extended factor existence theory using spectral conditions.

Abstract

Let $a\leq b$ be two positive integers. We say that a graph $G$ has all $[a,b]$-factors if it has an $h$-factor for every function $h: V(G)\rightarrow \mathbb{Z}^+$ such that $a\le h(v) \le b$ for all $v\in V(G)$ and $\sum_{v\in V(G)}h(v)\equiv 0\pmod 2$, and has all fractional $[a,b]$-factors if it has a fractional $p$-factor for every $p: V(G) \rightarrow \mathbb{Z}^+$ such that $a\le p(v)\le b$ for all $v\in V(G)$. In this paper, we provide tight spectral radius conditions for graphs having all $[a,b]$-factors ($3\leq a<b$) and all fractional $[a,b]$-factors ($1\leq a<b$), respectively.