Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$h: E(G) \rightarrow [0,1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>h</mi> <mo>:</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> be an edge indicator function of a graph G. For <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$k \in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>, a fractional k-factor in G is a spanning subgraph induced by <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$F_{h}=\{e \in E(G): h(e)>0\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>F</mi> <mi>h</mi> </msub> <mo>=</mo> <mrow> <mo stretchy="false">{</mo> <mi>e</mi> <mo>∈</mo> <mi>E</mi> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mi>h</mi> <mrow> <mo stretchy="false">(</mo> <mi>e</mi> <mo stretchy="false">)</mo> </mrow> <mo>></mo> <mn>0</mn> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> if <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\sum _{e \in E_{G}(v)}h(e)=k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>∑</mo> <mrow> <mi>e</mi> <mo>∈</mo> <msub> <mi>E</mi> <mi>G</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </msub> <mi>h</mi> <mrow> <mo stretchy="false">(</mo> <mi>e</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>k</mi> </mrow> </math></EquationSource> </InlineEquation> for every vertex <InlineEquation ID="IEq5"> <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>, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$E_{G}(v)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>E</mi> <mi>G</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denotes the set of edges incident with v in G. We say G has a fractional k-factor if such an edge indicator function h exists. A graph G is a fractional (k, l)-critical graph if, after deleting any l vertices from G, the resulting subgraph still admits a fractional k-factor. It is interesting to determine whether a graph is fractional (k, l)-critical. In this paper, we establish some sufficient conditions involving the size and the spectral radius to guarantee that a graph G with minimum degree <InlineEquation ID="IEq7"> <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> is fractional (k, l)-critical, which generalize the results on fractional k-factors due to Li et al. (Linear Algebra Appl 715:32–45, 2025).

Characterizations of fractional factor-critical graphs via size and spectral radius

期刊论文
发表日期：

2026年4月21日

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Xiaoyun Lv,
Jianxi Li,
Shou-Jun Xu

摘要

Let $h: E(G) \rightarrow [0,1]$ $h : E (G) \to [0, 1]$ be an edge indicator function of a graph G. For $k \in \mathbb {N}$ $k \in N$ , a fractional k-factor in G is a spanning subgraph induced by $F_{h}=\{e \in E(G): h(e)>0\}$ $F_{h} = {e \in E (G) : h (e) > 0}$ if $\sum _{e \in E_{G}(v)}h(e)=k$ $\sum_{e \in E_{G} (v)} h (e) = k$ for every vertex $v \in V(G)$ $v \in V (G)$ , where $E_{G}(v)$ $E_{G} (v)$ denotes the set of edges incident with v in G. We say G has a fractional k-factor if such an edge indicator function h exists. A graph G is a fractional (k, l)-critical graph if, after deleting any l vertices from G, the resulting subgraph still admits a fractional k-factor. It is interesting to determine whether a graph is fractional (k, l)-critical. In this paper, we establish some sufficient conditions involving the size and the spectral radius to guarantee that a graph G with minimum degree $\delta (G)$ $δ (G)$ is fractional (k, l)-critical, which generalize the results on fractional k-factors due to Li et al. (Linear Algebra Appl 715:32–45, 2025).