Sufficient conditions for a graph with minimum degree to be k-critical with respect to [1,b]-odd factor

TL;DR

This paper establishes size and spectral radius conditions under which a graph with a given minimum degree is guaranteed to be k-critical with respect to the existence of a [1,b]-odd factor, extending understanding of graph factors.

Contribution

It provides new sufficient conditions based on size and spectral radius for graphs to be k-critical regarding [1,b]-odd factors, a novel extension in graph factor theory.

Findings

Size conditions ensure k-criticality for graphs with minimum degree.

Spectral radius bounds guarantee the existence of [1,b]-odd factors.

Results apply to graphs of order at least k+2.

Abstract

A spanning subgraph $F$ of a graph $G$ is called a $[1,b]$-odd factor if $b\equiv1$ (mod 2) and $d_F(v)\in\{1,3,\ldots,b\}$ for every $v\in V(G)$. A graph $G$ of order $n\geq k+2$ is $k$-critical with respect to $[1,b]$-odd factor if for any $X\subseteq V(G)$ with $|X|=k$, $G-X$ has a $[1,b]$-odd factor. In this paper, we provide a size and spectral radius conditions for a graph with minimum degree to be $k$-critical with respect to $[1,b]$-odd factor, respectively.