图的复流数下限:几何方法

Pub Date : 2024-02-04 DOI:10.1002/jgt.23075
Davide Mattiolo, Giuseppe Mazzuoccolo, Jozef Rajník, Gloria Tabarelli
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A complex nowhere-zero <span></span><math>\n <semantics>\n <mrow>\n <mi>r</mi>\n </mrow>\n <annotation> $r$</annotation>\n </semantics></math>-flow on a graph <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> is an orientation of <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> together with an assignment <span></span><math>\n <semantics>\n <mrow>\n <mi>φ</mi>\n <mo>:</mo>\n <mi>E</mi>\n <mrow>\n <mo>(</mo>\n <mi>G</mi>\n <mo>)</mo>\n </mrow>\n <mo>→</mo>\n <mi>C</mi>\n </mrow>\n <annotation> $\\varphi :E(G)\\to {\\mathbb{C}}$</annotation>\n </semantics></math> such that, for all <span></span><math>\n <semantics>\n <mrow>\n <mi>e</mi>\n <mo>∈</mo>\n <mi>E</mi>\n <mrow>\n <mo>(</mo>\n <mi>G</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $e\\in E(G)$</annotation>\n </semantics></math>, the Euclidean norm of the complex number <span></span><math>\n <semantics>\n <mrow>\n <mi>φ</mi>\n <mrow>\n <mo>(</mo>\n <mi>e</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $\\varphi (e)$</annotation>\n </semantics></math> lies in the interval <span></span><math>\n <semantics>\n <mrow>\n <mo>[</mo>\n <mrow>\n <mn>1</mn>\n <mo>,</mo>\n <mi>r</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n <mo>]</mo>\n </mrow>\n <annotation> $[1,r-1]$</annotation>\n </semantics></math> and, for every vertex, the incoming flow is equal to the outgoing flow. The complex flow number of a bridgeless graph <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math>, denoted by <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>ϕ</mi>\n <mi>C</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>G</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> ${\\phi }_{{\\mathbb{C}}}(G)$</annotation>\n </semantics></math>, is the minimum of the real numbers <span></span><math>\n <semantics>\n <mrow>\n <mi>r</mi>\n </mrow>\n <annotation> $r$</annotation>\n </semantics></math> such that <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> admits a complex nowhere-zero <span></span><math>\n <semantics>\n <mrow>\n <mi>r</mi>\n </mrow>\n <annotation> $r$</annotation>\n </semantics></math>-flow. The exact computation of <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>ϕ</mi>\n <mi>C</mi>\n </msub>\n </mrow>\n <annotation> ${\\phi }_{{\\mathbb{C}}}$</annotation>\n </semantics></math> seems to be a hard task even for very small and symmetric graphs. In particular, the exact value of <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>ϕ</mi>\n <mi>C</mi>\n </msub>\n </mrow>\n <annotation> ${\\phi }_{{\\mathbb{C}}}$</annotation>\n </semantics></math> is known only for families of graphs where a lower bound can be trivially proved. Here, we use geometric and combinatorial arguments to give a nontrivial lower bound for <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>ϕ</mi>\n <mi>C</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>G</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> ${\\phi }_{{\\mathbb{C}}}(G)$</annotation>\n </semantics></math> in terms of the odd-girth of a cubic graph <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> (i.e., the length of a shortest odd cycle) and we show that this lower bound is tight. This result relies on the exact computation of the complex flow number of the wheel graph <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>W</mi>\n <mi>n</mi>\n </msub>\n </mrow>\n <annotation> ${W}_{n}$</annotation>\n </semantics></math>. In particular, we show that for every odd <span></span><math>\n <semantics>\n <mrow>\n <mi>n</mi>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math>, the value of <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>ϕ</mi>\n <mi>C</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <msub>\n <mi>W</mi>\n <mi>n</mi>\n </msub>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> ${\\phi }_{{\\mathbb{C}}}({W}_{n})$</annotation>\n </semantics></math> arises from one of three suitable configurations of points in the complex plane according to the congruence of <span></span><math>\n <semantics>\n <mrow>\n <mi>n</mi>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math> modulo 6.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A lower bound for the complex flow number of a graph: A geometric approach\",\"authors\":\"Davide Mattiolo,&nbsp;Giuseppe Mazzuoccolo,&nbsp;Jozef Rajník,&nbsp;Gloria Tabarelli\",\"doi\":\"10.1002/jgt.23075\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>r</mi>\\n <mo>≥</mo>\\n <mn>2</mn>\\n </mrow>\\n <annotation> $r\\\\ge 2$</annotation>\\n </semantics></math> be a real number. A complex nowhere-zero <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>r</mi>\\n </mrow>\\n <annotation> $r$</annotation>\\n </semantics></math>-flow on a graph <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math> is an orientation of <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math> together with an assignment <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>φ</mi>\\n <mo>:</mo>\\n <mi>E</mi>\\n <mrow>\\n <mo>(</mo>\\n <mi>G</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>→</mo>\\n <mi>C</mi>\\n </mrow>\\n <annotation> $\\\\varphi :E(G)\\\\to {\\\\mathbb{C}}$</annotation>\\n </semantics></math> such that, for all <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>e</mi>\\n <mo>∈</mo>\\n <mi>E</mi>\\n <mrow>\\n <mo>(</mo>\\n <mi>G</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $e\\\\in E(G)$</annotation>\\n </semantics></math>, the Euclidean norm of the complex number <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>φ</mi>\\n <mrow>\\n <mo>(</mo>\\n <mi>e</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\varphi (e)$</annotation>\\n </semantics></math> lies in the interval <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>[</mo>\\n <mrow>\\n <mn>1</mn>\\n <mo>,</mo>\\n <mi>r</mi>\\n <mo>−</mo>\\n <mn>1</mn>\\n </mrow>\\n <mo>]</mo>\\n </mrow>\\n <annotation> $[1,r-1]$</annotation>\\n </semantics></math> and, for every vertex, the incoming flow is equal to the outgoing flow. The complex flow number of a bridgeless graph <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math>, denoted by <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>ϕ</mi>\\n <mi>C</mi>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>G</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> ${\\\\phi }_{{\\\\mathbb{C}}}(G)$</annotation>\\n </semantics></math>, is the minimum of the real numbers <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>r</mi>\\n </mrow>\\n <annotation> $r$</annotation>\\n </semantics></math> such that <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math> admits a complex nowhere-zero <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>r</mi>\\n </mrow>\\n <annotation> $r$</annotation>\\n </semantics></math>-flow. The exact computation of <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>ϕ</mi>\\n <mi>C</mi>\\n </msub>\\n </mrow>\\n <annotation> ${\\\\phi }_{{\\\\mathbb{C}}}$</annotation>\\n </semantics></math> seems to be a hard task even for very small and symmetric graphs. In particular, the exact value of <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>ϕ</mi>\\n <mi>C</mi>\\n </msub>\\n </mrow>\\n <annotation> ${\\\\phi }_{{\\\\mathbb{C}}}$</annotation>\\n </semantics></math> is known only for families of graphs where a lower bound can be trivially proved. 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引用次数: 0

摘要

设 r ≥ 2 $r\ge 2$ 为实数。图 G $G$ 上的复数无处-零 r $r$ -流是 G $G$ 的一个取向以及一个赋值 φ : E ( G ) → C $\varphi :E(G)\to {\mathbb{C}}$ ,这样,对于所有 e∈ E ( G ) $e\in E(G)$ ,复数 φ ( e ) $\varphi (e)$ 的欧氏规范位于区间 [ 1 , r - 1 ]。 $[1,r-1]$ 并且,对于每个顶点,流入流量等于流出流量。无桥图 G $G$ 的复流数用 ϕ C ( G ) ${\phi }_{{\mathbb{C}}}(G)$ 表示,是实数 r $r$ 中的最小值,使得 G $G$ 可以容纳无处为零的复 r $r$ 流。即使对于非常小的对称图,精确计算 ϕ C ${\phi }_{\mathbb{C}}$ 似乎也是一项艰巨的任务。特别是,j C ${\phi }_{\mathbb{C}}$的精确值只有在可以微不足道地证明下界的图族中才是已知的。在这里,我们利用几何和组合论证,以立方图 G $G$ 的奇数周长(即最短奇数周期的长度)为单位,给出了 ϕ C ( G ) ${\phi }_{\mathbb{C}}(G)$ 的非微不足道的下界,并证明这个下界是严密的。这一结果依赖于车轮图 W n ${W}_{n}$ 复流数的精确计算。特别是,我们证明了对于每一个奇数 n $n$ ,ϕ C ( W n ) ${\phi }_{{\mathbb{C}}}({W}_{n})$ 的值产生于复数平面中根据 n $n$ modulo 6 的同余式的三个合适配置点之一。
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A lower bound for the complex flow number of a graph: A geometric approach

Let r 2 $r\ge 2$ be a real number. A complex nowhere-zero r $r$ -flow on a graph G $G$ is an orientation of G $G$ together with an assignment φ : E ( G ) C $\varphi :E(G)\to {\mathbb{C}}$ such that, for all e E ( G ) $e\in E(G)$ , the Euclidean norm of the complex number φ ( e ) $\varphi (e)$ lies in the interval [ 1 , r 1 ] $[1,r-1]$ and, for every vertex, the incoming flow is equal to the outgoing flow. The complex flow number of a bridgeless graph G $G$ , denoted by ϕ C ( G ) ${\phi }_{{\mathbb{C}}}(G)$ , is the minimum of the real numbers r $r$ such that G $G$ admits a complex nowhere-zero r $r$ -flow. The exact computation of ϕ C ${\phi }_{{\mathbb{C}}}$ seems to be a hard task even for very small and symmetric graphs. In particular, the exact value of ϕ C ${\phi }_{{\mathbb{C}}}$ is known only for families of graphs where a lower bound can be trivially proved. Here, we use geometric and combinatorial arguments to give a nontrivial lower bound for ϕ C ( G ) ${\phi }_{{\mathbb{C}}}(G)$ in terms of the odd-girth of a cubic graph G $G$ (i.e., the length of a shortest odd cycle) and we show that this lower bound is tight. This result relies on the exact computation of the complex flow number of the wheel graph W n ${W}_{n}$ . In particular, we show that for every odd n $n$ , the value of ϕ C ( W n ) ${\phi }_{{\mathbb{C}}}({W}_{n})$ arises from one of three suitable configurations of points in the complex plane according to the congruence of n $n$ modulo 6.

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