Spectral radius, fractional $[a,b]$-factor and ID-factor-critical graphs

TL;DR

This paper establishes spectral radius-based conditions that guarantee the existence of fractional $[a,b]$-factors and ID-factor-criticality in graphs, extending previous results and providing tight bounds.

Contribution

It introduces new spectral radius criteria for fractional $[a,b]$-factors and ID-factor-critical graphs, expanding the theoretical understanding of graph factors.

Findings

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

Established spectral radius bounds ensuring ID-factor-criticality.

Extended previous results by Wei and Zhang on spectral conditions.

Abstract

Let $G$ be a graph and $h: E(G)\rightarrow [0,1]$ be a function. For any two positive integers $a$ and $b$ with $a\leq b$, a fractional $[a,b]$-factor of $G$ with the indicator function $h$ is a spanning subgraph with vertex set $V(G)$ and edge set $E_h$ such that $a\leq\sum_{e\in E_{G}(v)}h(e)\leq b$ for any vertex $v\in V(G)$, where $E_h = \{e\in E(G)|h(e)>0\}$ and $E_{G}(v)=\{e\in E(G)| e~\mbox{is incident with}~v~\mbox{in}~G\}$. A graph $G$ is ID-factor-critical if for every independent set $I$ of $G$ whose size has the same parity as $|V(G)|$, $G-I$ has a perfect matching. In this paper, we present a tight sufficient condition based on the spectral radius for a graph to contain a fractional $[a,b]$-factor, which extends the result of Wei and Zhang [Discrete Math. 346 (2023) 113269]. Furthermore, we also prove a tight sufficient condition in terms of the spectral radius for a graph with minimum degree $\delta$ to be ID-factor-critical.