Let a,  b,  k be integers with <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$1 \le a \le b$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>a</mi> <mo>≤</mo> <mi>b</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$k \ge 0.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>0</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> A graph G is all (fractional) (a, b, k)-critical if after deleting any k vertices of G the remaining graph of G has all (fractional) [a, b]-factors. If <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$k=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, then the graph G has all (fractional) [a, b]-factors. Hence all (fractional) (a, b, k)-critical graphs are natural generalization of having all (fractional) [a, b]-factors in G. For <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$1 \le a \le b$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>a</mi> <mo>≤</mo> <mi>b</mi> </mrow> </math></EquationSource> </InlineEquation>, by using technical structure theorems and typical spectral methods, we in this paper provide tight sufficient conditions in terms of the spectral radius for a graph to be all (a, b, k)-critical and all fractional (a, b, k)-critical, respectively. Our results extend and improve the corresponding results of Zheng, Wang and Huang on all [a, b]-factors and all fractional [a, b]-factors with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$a < b$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo><</mo> <mi>b</mi> </mrow> </math></EquationSource> </InlineEquation> in [Discrete Math. 347 (2024) 113975], and strengthen the results of Wei and Zhang [Discrete Math. 346 (2023) 113269] and Fan, Lin and Lu [Discrete Math. 345 (2022) 112892] on [a, b]-factors.

Spectral extremal problem on all (a, b, k)-critical graphs

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发表日期：

2025年9月11日

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Qixuan Yuan,
Ruifang Liu,
Jinjiang Yuan

摘要

Let a, b, k be integers with $1 \le a \le b$ $1 \leq a \leq b$ and $k \ge 0.$ $k \geq 0 .$ A graph G is all (fractional) (a, b, k)-critical if after deleting any k vertices of G the remaining graph of G has all (fractional) [a, b]-factors. If $k=0$ $k = 0$ , then the graph G has all (fractional) [a, b]-factors. Hence all (fractional) (a, b, k)-critical graphs are natural generalization of having all (fractional) [a, b]-factors in G. For $1 \le a \le b$ $1 \leq a \leq b$ , by using technical structure theorems and typical spectral methods, we in this paper provide tight sufficient conditions in terms of the spectral radius for a graph to be all (a, b, k)-critical and all fractional (a, b, k)-critical, respectively. Our results extend and improve the corresponding results of Zheng, Wang and Huang on all [a, b]-factors and all fractional [a, b]-factors with $a < b$ $a < b$ in [Discrete Math. 347 (2024) 113975], and strengthen the results of Wei and Zhang [Discrete Math. 346 (2023) 113269] and Fan, Lin and Lu [Discrete Math. 345 (2022) 112892] on [a, b]-factors.