计算图的连接分区

Pub Date : 2024-06-02 DOI:10.1002/jgt.23127

Yair Caro, Balázs Patkós, Zsolt Tuza, Máté Vizer

{"title":"计算图的连接分区","authors":"Yair Caro, Balázs Patkós, Zsolt Tuza, Máté Vizer","doi":"10.1002/jgt.23127","DOIUrl":null,"url":null,"abstract":"Motivated by the theorem of Győri and Lovász, we consider the following problem. For a connected graph <math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> on <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math> vertices and <math>\n <semantics>\n <mrow>\n <mi>m</mi>\n </mrow>\n <annotation> $m$</annotation>\n </semantics></math> edges determine the number <math>\n <semantics>\n <mrow>\n <mi>P</mi>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>G</mi>\n <mo>,</mo>\n <mi>k</mi>\n </mrow>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $P(G,k)$</annotation>\n </semantics></math> of unordered solutions of positive integers <math>\n <semantics>\n <mrow>\n <msubsup>\n <mo>∑</mo>\n <mrow>\n <mi>i</mi>\n <mo>=</mo>\n <mn>1</mn>\n </mrow>\n <mi>k</mi>\n </msubsup>\n <msub>\n <mi>m</mi>\n <mi>i</mi>\n </msub>\n <mo>=</mo>\n <mi>m</mi>\n </mrow>\n <annotation> ${\\sum }_{i=1}^{k}{m}_{i}=m$</annotation>\n </semantics></math> such that every <math>\n <semantics>\n <mrow>\n <msub>\n <mi>m</mi>\n <mi>i</mi>\n </msub>\n </mrow>\n <annotation> ${m}_{i}$</annotation>\n </semantics></math> is realized by a connected subgraph <math>\n <semantics>\n <mrow>\n <msub>\n <mi>H</mi>\n <mi>i</mi>\n </msub>\n </mrow>\n <annotation> ${H}_{i}$</annotation>\n </semantics></math> of <math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> with <math>\n <semantics>\n <mrow>\n <msub>\n <mi>m</mi>\n <mi>i</mi>\n </msub>\n </mrow>\n <annotation> ${m}_{i}$</annotation>\n </semantics></math> edges. We also consider the vertex-partition analogue. We prove various lower bounds on <math>\n <semantics>\n <mrow>\n <mi>P</mi>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>G</mi>\n <mo>,</mo>\n <mi>k</mi>\n </mrow>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $P(G,k)$</annotation>\n </semantics></math> as a function of the number <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math> of vertices in <math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math>, as a function of the average degree <math>\n <semantics>\n <mrow>\n <mi>d</mi>\n </mrow>\n <annotation> $d$</annotation>\n </semantics></math> of <math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math>, and also as the size <math>\n <semantics>\n <mrow>\n <mi>C</mi>\n <mi>M</mi>\n <msub>\n <mi>C</mi>\n <mi>r</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>G</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $CM{C}_{r}(G)$</annotation>\n </semantics></math> of <math>\n <semantics>\n <mrow>\n <mi>r</mi>\n </mrow>\n <annotation> $r$</annotation>\n </semantics></math>-partite connected maximum cuts of <math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math>. Those three lower bounds are tight up to a multiplicative constant. We also prove that the number <math>\n <semantics>\n <mrow>\n <mi>π</mi>\n <mrow>\n <mo>(</mo>\n <mrow>\n <mi>G</mi>\n <mo>,</mo>\n <mi>k</mi>\n </mrow>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> $\\pi (G,k)$</annotation>\n </semantics></math> of unordered <math>\n <semantics>\n <mrow>\n <mi>k</mi>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>-tuples with <math>\n <semantics>\n <mrow>\n <msubsup>\n <mo>∑</mo>\n <mrow>\n <mi>i</mi>\n <mo>=</mo>\n <mn>1</mn>\n </mrow>\n <mi>k</mi>\n </msubsup>\n <msub>\n <mi>n</mi>\n <mi>i</mi>\n </msub>\n <mo>=</mo>\n <mi>n</mi>\n </mrow>\n <annotation> ${\\sum }_{i=1}^{k}{n}_{i}=n$</annotation>\n </semantics></math>, that are realizable by vertex partitions into <math>\n <semantics>\n <mrow>\n <mi>k</mi>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math> connected parts, is at least <math>\n <semantics>\n <mrow>\n <mi>Ω</mi>\n <mrow>\n <mo>(</mo>\n <msup>\n <mi>d</mi>\n <mrow>\n <mi>k</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </msup>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation> ${\\rm{\\Omega }}({d}^{k-1})$</annotation>\n </semantics></math>.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Counting connected partitions of graphs\",\"authors\":\"Yair Caro, Balázs Patkós, Zsolt Tuza, Máté Vizer\",\"doi\":\"10.1002/jgt.23127\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Motivated by the theorem of Győri and Lovász, we consider the following problem. For a connected graph <math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math> on <math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n <annotation> $n$</annotation>\\n </semantics></math> vertices and <math>\\n <semantics>\\n <mrow>\\n <mi>m</mi>\\n </mrow>\\n <annotation> $m$</annotation>\\n </semantics></math> edges determine the number <math>\\n <semantics>\\n <mrow>\\n <mi>P</mi>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mi>G</mi>\\n <mo>,</mo>\\n <mi>k</mi>\\n </mrow>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $P(G,k)$</annotation>\\n </semantics></math> of unordered solutions of positive integers <math>\\n <semantics>\\n <mrow>\\n <msubsup>\\n <mo>∑</mo>\\n <mrow>\\n <mi>i</mi>\\n <mo>=</mo>\\n <mn>1</mn>\\n </mrow>\\n <mi>k</mi>\\n </msubsup>\\n <msub>\\n <mi>m</mi>\\n <mi>i</mi>\\n </msub>\\n <mo>=</mo>\\n <mi>m</mi>\\n </mrow>\\n <annotation> ${\\\\sum }_{i=1}^{k}{m}_{i}=m$</annotation>\\n </semantics></math> such that every <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>m</mi>\\n <mi>i</mi>\\n </msub>\\n </mrow>\\n <annotation> ${m}_{i}$</annotation>\\n </semantics></math> is realized by a connected subgraph <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>H</mi>\\n <mi>i</mi>\\n </msub>\\n </mrow>\\n <annotation> ${H}_{i}$</annotation>\\n </semantics></math> of <math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math> with <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>m</mi>\\n <mi>i</mi>\\n </msub>\\n </mrow>\\n <annotation> ${m}_{i}$</annotation>\\n </semantics></math> edges. We also consider the vertex-partition analogue. We prove various lower bounds on <math>\\n <semantics>\\n <mrow>\\n <mi>P</mi>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mi>G</mi>\\n <mo>,</mo>\\n <mi>k</mi>\\n </mrow>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $P(G,k)$</annotation>\\n </semantics></math> as a function of the number <math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n <annotation> $n$</annotation>\\n </semantics></math> of vertices in <math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math>, as a function of the average degree <math>\\n <semantics>\\n <mrow>\\n <mi>d</mi>\\n </mrow>\\n <annotation> $d$</annotation>\\n </semantics></math> of <math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math>, and also as the size <math>\\n <semantics>\\n <mrow>\\n <mi>C</mi>\\n <mi>M</mi>\\n <msub>\\n <mi>C</mi>\\n <mi>r</mi>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>G</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $CM{C}_{r}(G)$</annotation>\\n </semantics></math> of <math>\\n <semantics>\\n <mrow>\\n <mi>r</mi>\\n </mrow>\\n <annotation> $r$</annotation>\\n </semantics></math>-partite connected maximum cuts of <math>\\n <semantics>\\n <mrow>\\n <mi>G</mi>\\n </mrow>\\n <annotation> $G$</annotation>\\n </semantics></math>. Those three lower bounds are tight up to a multiplicative constant. We also prove that the number <math>\\n <semantics>\\n <mrow>\\n <mi>π</mi>\\n <mrow>\\n <mo>(</mo>\\n <mrow>\\n <mi>G</mi>\\n <mo>,</mo>\\n <mi>k</mi>\\n </mrow>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\pi (G,k)$</annotation>\\n </semantics></math> of unordered <math>\\n <semantics>\\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n <annotation> $k$</annotation>\\n </semantics></math>-tuples with <math>\\n <semantics>\\n <mrow>\\n <msubsup>\\n <mo>∑</mo>\\n <mrow>\\n <mi>i</mi>\\n <mo>=</mo>\\n <mn>1</mn>\\n </mrow>\\n <mi>k</mi>\\n </msubsup>\\n <msub>\\n <mi>n</mi>\\n <mi>i</mi>\\n </msub>\\n <mo>=</mo>\\n <mi>n</mi>\\n </mrow>\\n <annotation> ${\\\\sum }_{i=1}^{k}{n}_{i}=n$</annotation>\\n </semantics></math>, that are realizable by vertex partitions into <math>\\n <semantics>\\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n <annotation> $k$</annotation>\\n </semantics></math> connected parts, is at least <math>\\n <semantics>\\n <mrow>\\n <mi>Ω</mi>\\n <mrow>\\n <mo>(</mo>\\n <msup>\\n <mi>d</mi>\\n <mrow>\\n <mi>k</mi>\\n <mo>−</mo>\\n <mn>1</mn>\\n </mrow>\\n </msup>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> ${\\\\rm{\\\\Omega }}({d}^{k-1})$</annotation>\\n </semantics></math>.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-06-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23127\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23127","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 0

摘要

受 Győri 和 Lovász 的定理启发，我们考虑了以下问题。对于 n $n$ 个顶点和 m $m$ 条边的连通图 G $G$ 确定正整数无序解的个数 P ( G , k ) $P(G,k)$ ∑ i = 1 k m i = m ${\sum }_{i=1}^{k}{m}_{i}=m$ ，使得每个 m i ${m}_{i}$ 都由 G $G$ 的连通子图 H i ${H}_{i}$ 和 m i ${m}_{i}$ 条边实现。我们还考虑了顶点分区的类似方法。我们证明了 P ( G , k ) $P(G,k)$作为 G $G$ 中顶点数 n $n$ 的函数、G $G$ 的平均度 d $d$ 的函数以及 G $G$ 的 r $r$ 部分连通最大切分的大小 C M C r ( G ) $CM{C}_{r}(G)$的各种下界。这三个下界都很紧，直到一个乘法常数。我们还证明，∑ i = 1 k n i = n ${sum }_{i=1}^{k}{n}_{i}=n$ 的无序 k $k$ 图元的π ( G , k ) $\pi (G,k)$ 数目至少为 Ω ( d k - 1 ) ${\rm{\Omega }}({d}^{k-1})$ 。

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Counting connected partitions of graphs

Motivated by the theorem of Győri and Lovász, we consider the following problem. For a connected graph $G$ on $n$ vertices and $m$ edges determine the number $P (G, k)$ of unordered solutions of positive integers $\sum_{i = 1}^{k} m_{i} = m$ such that every $m_{i}$ is realized by a connected subgraph $H_{i}$ of $G$ with $m_{i}$ edges. We also consider the vertex-partition analogue. We prove various lower bounds on $P (G, k)$ as a function of the number $n$ of vertices in $G$ , as a function of the average degree $d$ of $G$ , and also as the size $C M C_{r} (G)$ of $r$ -partite connected maximum cuts of $G$ . Those three lower bounds are tight up to a multiplicative constant. We also prove that the number $π (G, k)$ of unordered $k$ -tuples with $\sum_{i = 1}^{k} n_{i} = n$ , that are realizable by vertex partitions into $k$ connected parts, is at least $Ω (d^{k - 1})$ .

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