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

Abstract

Let $G$ be a graph and $h:E(G)\to [0,1]$ be a function. For any two positive integers $a$ and $b$ with $a\le b$, 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 $a\le {\sum }_{e\in E_G(v)}h(e)\le b$ 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.

14 pages, 2 figures