Counting connected partitions of graphs

IF 0.9 3区 数学 Q2 MATHEMATICS
Yair Caro, Balázs Patkós, Zsolt Tuza, Máté Vizer
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For a connected graph <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> on <span></span><math>\n <semantics>\n <mrow>\n <mi>n</mi>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math> vertices and <span></span><math>\n <semantics>\n <mrow>\n <mi>m</mi>\n </mrow>\n <annotation> $m$</annotation>\n </semantics></math> edges determine the number <span></span><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 <span></span><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 <span></span><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 <span></span><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 <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> with <span></span><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 <span></span><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 <span></span><math>\n <semantics>\n <mrow>\n <mi>n</mi>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math> of vertices in <span></span><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 <span></span><math>\n <semantics>\n <mrow>\n <mi>d</mi>\n </mrow>\n <annotation> $d$</annotation>\n </semantics></math> of <span></span><math>\n <semantics>\n <mrow>\n <mi>G</mi>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math>, and also as the size <span></span><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 <span></span><math>\n <semantics>\n <mrow>\n <mi>r</mi>\n </mrow>\n <annotation> $r$</annotation>\n </semantics></math>-partite connected maximum cuts of <span></span><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 <span></span><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 <span></span><math>\n <semantics>\n <mrow>\n <mi>k</mi>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math>-tuples with <span></span><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 <span></span><math>\n <semantics>\n <mrow>\n <mi>k</mi>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math> connected parts, is at least <span></span><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>.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 2","pages":"381-392"},"PeriodicalIF":0.9000,"publicationDate":"2024-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23127","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

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

计算图的连接分区
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 ) ${\rm{\Omega }}({d}^{k-1})$ 。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Journal of Graph Theory
Journal of Graph Theory 数学-数学
CiteScore
1.60
自引率
22.20%
发文量
130
审稿时长
6-12 weeks
期刊介绍: The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .
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