Some sufficient conditions for a graph with minimum degree to be $k$-critical with respect to $[1,b]$-odd factors

TL;DR

This paper establishes sufficient spectral radius conditions related to distance matrices that ensure a graph with a given minimum degree is $k$-critical concerning $[1,b]$-odd factors, extending understanding of graph factorization properties.

Contribution

It introduces new spectral radius criteria based on distance matrices to determine $k$-criticality for graphs with minimum degree, focusing on $[1,b]$-odd factors.

Findings

Spectral radius conditions guarantee $k$-criticality

Results apply to graphs with specified minimum degree

Provides bounds based on distance spectral parameters

Abstract

A graph $G$ is $k$-factor-critical if $G-S$ has a perfect matching for every subset $S \subseteq V(G)$ with $|S|=k$. A spanning subgraph $H$ of $G$ is called a $[1,b]$-odd factor if $b \equiv 1 \pmod{2}$ and $d_{H}(v) \in\left\lbrace 1, 3, \ldots, b\right\rbrace$ for every $v\in V(G),$ where $d_{H}(v)$ denotes the degree of vertex $v$ in $H$. Moreover, $G$ is said to be $k$-critical with respect to $[1,b]$-odd factors if $G-X$ contains a $[1,b]$-odd factor for every subset $X \subseteq V(G)$ with $|X|=k$. In this paper, we provide some sufficient conditions based on the distance spectral radius and the distance signless Laplacian spectral radius for a graph with minimum degree to be $k$-critical with respect to $[1,b]$-odd factors.