A graph G has the k-strong parity property if for any <InlineEquation ID="IEq1"> <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 |X| even, G contains a spanning subgraph F with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$d_F(u)\equiv 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>d</mi> <mi>F</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>≡</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> (mod 2) for each <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$u\in X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>∈</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$d_F(v)\in \{k,k+2,k+4,\ldots \}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>d</mi> <mi>F</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <mrow> <mo stretchy="false">{</mo> <mi>k</mi> <mo>,</mo> <mi>k</mi> <mo>+</mo> <mn>2</mn> <mo>,</mo> <mi>k</mi> <mo>+</mo> <mn>4</mn> <mo>,</mo> <mo>…</mo> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for each <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$v\in V(G)\setminus X$</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> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$k\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> is an even integer. Kano and Matsumura proposed a characterization for a graph with the k-strong parity property (Kano and Matsumura in Graphs Combin 41:55, 2025). In this paper, we first give a size condition for a graph to have the k-strong parity property. Then we establish a signless Laplacian spectral radius condition to guarantee that a graph has the k-strong parity property.

Some results on the k-strong parity property in a graph

期刊论文
发表日期：

2025年12月18日

查看全文

Jie Wu

摘要

A graph G has the k-strong parity property if for any $X\subseteq V(G)$ $X \subseteq V (G)$ with |X| even, G contains a spanning subgraph F with $d_F(u)\equiv 1$ $d_{F} (u) \equiv 1$ (mod 2) for each $u\in X$ $u \in X$ and $d_F(v)\in \{k,k+2,k+4,\ldots \}$ $d_{F} (v) \in {k, k + 2, k + 4, \dots}$ for each $v\in V(G)\setminus X$ $v \in V (G) \ X$ , where $k\ge 2$ $k \geq 2$ is an even integer. Kano and Matsumura proposed a characterization for a graph with the k-strong parity property (Kano and Matsumura in Graphs Combin 41:55, 2025). In this paper, we first give a size condition for a graph to have the k-strong parity property. Then we establish a signless Laplacian spectral radius condition to guarantee that a graph has the k-strong parity property.