An [a, b]-factor of a graph G is a spanning subgraph H satisfying <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$a\le d_{H}(v)\le b$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>≤</mo> <msub> <mi>d</mi> <mi>H</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <mi>b</mi> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$v\in V(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>v</mi> <mo>∈</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. A graph G is termed an (a, b; k)-critical graph if every subgraph obtained by deleting any k vertices retains an [a, b]-factor, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$b\ge a\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>≥</mo> <mi>a</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$k\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. This concept generalizes the classical existence of [a, b]-factors to scenarios requiring robustness under vertex deletions. In this paper, we establish tight sufficient conditions involving size and spectral radius for a graph G with minimum degree <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\delta (G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> to be an (a, b; k)-critical graph, which generalize Fan et al. (A note on the spectral radius and [a, b]-factor of graphs, 2025). As corollaries, we also derive analogous conditions for fractional (a, b; k)-critical graphs.

Tight spectral and size conditions for (fractional) (a, b; k)-critical graphs

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发表日期：

2026年3月24日

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Yanhua Zhao,
Jinlong Cao,
Lidan Wang

摘要

An [a, b]-factor of a graph G is a spanning subgraph H satisfying $a\le d_{H}(v)\le b$ $a \leq d_{H} (v) \leq b$ for all $v\in V(G)$ $v \in V (G)$ . A graph G is termed an (a, b; k)-critical graph if every subgraph obtained by deleting any k vertices retains an [a, b]-factor, where $b\ge a\ge 1$ $b \geq a \geq 1$ and $k\ge 0$ $k \geq 0$ . This concept generalizes the classical existence of [a, b]-factors to scenarios requiring robustness under vertex deletions. In this paper, we establish tight sufficient conditions involving size and spectral radius for a graph G with minimum degree $\delta (G)$ $δ (G)$ to be an (a, b; k)-critical graph, which generalize Fan et al. (A note on the spectral radius and [a, b]-factor of graphs, 2025). As corollaries, we also derive analogous conditions for fractional (a, b; k)-critical graphs.