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

TL;DR

The paper addresses when a graph with minimum degree $\delta$ is $k$-critical with respect to the $[1,b]$-odd factor. It develops both a size-based criterion (Theorem $1.1$) and a spectral-radius criterion (Theorem $1.2$) for $(k+1)$-connected graphs of order $n$ with $n\equiv k\pmod{2}$ and odd $b$, identifying the extremal join graph $K_{\delta}\vee(K_{n-(b+1)\delta+bk-1}\cup(b\delta-bk+1)K_1)$ as the tight case. The proofs combine the Kano–Matsuda characterization of $k$-criticality with edge-count comparisons and eigenvalue interlacing via equitable partitions, handling three regimes for the cut-set size $s$. Together, the results extend factor-critical theory to $[1,b]$-odd factors under minimum-degree constraints and provide concrete, extremal-structure–based criteria with implications for graph factor robustness.

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.