A graph G is k-factor-critical if <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$G - S$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>-</mo> <mi>S</mi> </mrow> </math></EquationSource> </InlineEquation> has a perfect matching for every subset <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$S \subseteq V(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo>⊆</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$|S| = k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>S</mi> <mo stretchy="false">|</mo> <mo>=</mo> <mi>k</mi> </mrow> </math></EquationSource> </InlineEquation>. A spanning subgraph H of G is an [a, b]-factor if <InlineEquation ID="IEq4"> <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> holds 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 a and b are constants. Moreover, G is said to be k-critical with respect to a [1, b]-odd factor if <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$G - X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>-</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> contains a [1, b]-odd factor for every subset <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$X \subseteq V(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>⊆</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$|X| = k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">|</mo> <mi>X</mi> <mo stretchy="false">|</mo> <mo>=</mo> <mi>k</mi> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we provide a sufficient condition involving the signless Laplacian spectral radius to guarantee that G is k-critical with respect to a [1, b]-odd factor.

Signless Laplacian spectral conditions for k-critical graphs with respect to [1, b]-odd factors

期刊论文
发表日期：

2025年10月31日

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Tao Wang,
Mengke Yang,
Xiaojing Yang

摘要

A graph G is k-factor-critical if $G - S$ $G - S$ has a perfect matching for every subset $S \subseteq V(G)$ $S \subseteq V (G)$ with $|S| = k$ $| S | = k$ . A spanning subgraph H of G is an [a, b]-factor if $a \le d_H(v) \le b$ $a \leq d_{H} (v) \leq b$ holds for every vertex $v \in V(G)$ $v \in V (G)$ , where a and b are constants. Moreover, G is said to be k-critical with respect to a [1, b]-odd factor if $G - X$ $G - X$ contains a [1, b]-odd factor for every subset $X \subseteq V(G)$ $X \subseteq V (G)$ with $|X| = k$ $| X | = k$ . In this paper, we provide a sufficient condition involving the signless Laplacian spectral radius to guarantee that G is k-critical with respect to a [1, b]-odd factor.