Coloring hypergraphs with excluded minors

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics Pub Date : 2024-04-18 DOI:10.1016/j.ejc.2024.103971

Raphael Steiner

{"title":"Coloring hypergraphs with excluded minors","authors":"Raphael Steiner","doi":"10.1016/j.ejc.2024.103971","DOIUrl":null,"url":null,"abstract":"<div>Hadwiger’s conjecture, among the most famous open problems in graph theory, states that every graph that does not contain <math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math> as a minor is properly <math><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math>-colorable.The purpose of this work is to demonstrate that a natural extension of Hadwiger’s problem to hypergraph coloring exists, and to derive some first partial results and applications.Generalizing ordinary graph minors to hypergraphs, we say that a hypergraph <math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub></math> is a minor of a hypergraph <math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub></math>, if a hypergraph isomorphic to <math><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub></math> can be obtained from <math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub></math> via a finite sequence of the following operations:• deleting vertices and hyperedges,• contracting a hyperedge (i.e., merging the vertices of the hyperedge into a single vertex).First we show that a weak extension of Hadwiger’s conjecture to hypergraphs holds true: For every <math><mrow><mi>t</mi><mo>≥</mo><mn>1</mn></mrow></math>, there exists a finite (smallest) integer <math><mrow><mi>h</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></math> such that every hypergraph with no <math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math>-minor is <math><mrow><mi>h</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></math>-colorable, and we prove <math><mrow><mfenced><mrow><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow></mfenced><mo>≤</mo><mi>h</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>≤</mo><mn>2</mn><mi>g</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></math> where <math><mrow><mi>g</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></math> denotes the maximum chromatic number of graphs with no <math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math>-minor. Using the recent result by Delcourt and Postle that <math><mrow><mi>g</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>O</mi><mrow><mo>(</mo><mi>t</mi><mo>log</mo><mo>log</mo><mi>t</mi><mo>)</mo></mrow></mrow></math>, this yields <math><mrow><mi>h</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>O</mi><mrow><mo>(</mo><mi>t</mi><mo>log</mo><mo>log</mo><mi>t</mi><mo>)</mo></mrow></mrow></math>.We further conjecture that <math><mrow><mi>h</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mfenced><mrow><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow></mfenced></mrow></math>, i.e., that every hypergraph with no <math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math>-minor is <math><mfenced><mrow><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow></mfenced></math>-colorable for all <math><mrow><mi>t</mi><mo>≥</mo><mn>1</mn></mrow></math>, and prove this conjecture for all hypergraphs with independence number at most 2.By considering special classes of hypergraphs, the above additionally has some interesting applications for ordinary graph coloring, such as:• every graph <math><mi>G</mi></math> is <math><mrow><mi>O</mi><mrow><mo>(</mo><mi>k</mi><mi>t</mi><mo>log</mo><mo>log</mo><mi>t</mi><mo>)</mo></mrow></mrow></math>-colorable or contains a <math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math>-minor model all whose branch-sets are <math><mi>k</mi></math>-edge-connected,• every graph <math><mi>G</mi></math> is <math><mrow><mi>O</mi><mrow><mo>(</mo><mi>q</mi><mi>t</mi><mo>log</mo><mo>log</mo><mi>t</mi><mo>)</mo></mrow></mrow></math>-colorable or contains a <math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math>-minor model all whose branch-sets are modulo-<math><mi>q</mi></math>-connected (i.e., every pair of vertices in the same branch-set has a connecting path of prescribed length modulo <math><mi>q</mi></math>),• by considering cycle hypergraphs of digraphs, we obtain known results on strong minors in digraphs with large dichromatic number as special cases. We also construct digraphs with dichromatic number <math><mfenced><mrow><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow></mfenced></math> not containing the complete digraph on <math><mi>t</mi></math> vertices as a strong minor, thus answering a question by Mészáros and the author in the negative.</div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824000568/pdfft?md5=7ddba04d4bd02c12e555b22107b8bb39&pid=1-s2.0-S0195669824000568-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"European Journal of Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0195669824000568","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

Hadwiger’s conjecture, among the most famous open problems in graph theory, states that every graph that does not contain $K_{t}$ as a minor is properly $(t - 1)$ -colorable.

The purpose of this work is to demonstrate that a natural extension of Hadwiger’s problem to hypergraph coloring exists, and to derive some first partial results and applications.

Generalizing ordinary graph minors to hypergraphs, we say that a hypergraph $H_{1}$ is a minor of a hypergraph $H_{2}$ , if a hypergraph isomorphic to $H_{1}$ can be obtained from $H_{2}$ via a finite sequence of the following operations:

• deleting vertices and hyperedges,

• contracting a hyperedge (i.e., merging the vertices of the hyperedge into a single vertex).

First we show that a weak extension of Hadwiger’s conjecture to hypergraphs holds true: For every $t \geq 1$ , there exists a finite (smallest) integer $h (t)$ such that every hypergraph with no $K_{t}$ -minor is $h (t)$ -colorable, and we prove $(\frac{3}{2} (t - 1)) \leq h (t) \leq 2 g (t)$ where $g (t)$ denotes the maximum chromatic number of graphs with no $K_{t}$ -minor. Using the recent result by Delcourt and Postle that $g (t) = O (t log log t)$ , this yields $h (t) = O (t log log t)$ .

We further conjecture that $h (t) = (\frac{3}{2} (t - 1))$ , i.e., that every hypergraph with no $K_{t}$ -minor is $(\frac{3}{2} (t - 1))$ -colorable for all $t \geq 1$ , and prove this conjecture for all hypergraphs with independence number at most 2.

By considering special classes of hypergraphs, the above additionally has some interesting applications for ordinary graph coloring, such as:

• every graph $G$ is $O (k t log log t)$ -colorable or contains a $K_{t}$ -minor model all whose branch-sets are $k$ -edge-connected,

• every graph $G$ is $O (q t log log t)$ -colorable or contains a $K_{t}$ -minor model all whose branch-sets are modulo- $q$ -connected (i.e., every pair of vertices in the same branch-set has a connecting path of prescribed length modulo $q$ ),

• by considering cycle hypergraphs of digraphs, we obtain known results on strong minors in digraphs with large dichromatic number as special cases. We also construct digraphs with dichromatic number $(\frac{3}{2} (t - 1))$ not containing the complete digraph on $t$ vertices as a strong minor, thus answering a question by Mészáros and the author in the negative.

查看原文本刊更多论文

用排除最小值的超图着色

哈德维格猜想是图论中最著名的悬而未决问题之一，它指出每个不包含 Kt 作为次要图的图都是可着色的（t-1）图。本研究的目的是证明哈德维格问题在超图着色方面的自然扩展，并推导出一些初步的部分结果和应用。将普通图的次要图推广到超图，如果通过以下操作的有限序列可以从 H2 得到与 H1 同构的超图，我们就说超图 H1 是超图 H2 的次要图：- 删除顶点和超边，- 收缩一个超边（即首先，我们要证明哈德维格猜想在超图中的弱扩展是成立的：对于每个 t≥1，都存在一个有限（最小）整数 h(t)，使得每个没有 Kt-minor的超图都是 h(t)-colorable 的，我们证明 32(t-1)≤h(t)≤2g(t) 其中 g(t) 表示没有 Kt-minor 的图的最大色度数。我们进一步猜想，h(t)=32(t-1)，也就是说，每一个没有 Kt-minor的超图都有 Kt-minor、我们进一步猜想 h(t)=32(t-1)，即对于所有 t≥1 的超图，每一个没有 Kt-minor的超图都是 32(t-1)-colorable 的，并证明了所有独立数最多为 2 的超图的这一猜想。通过考虑特殊类别的超图，上述猜想对于普通图着色也有一些有趣的应用，例如：- 每个图 G 都是 O(ktloglogt)-colorable 或包含一个 Kt-minor 模型，其所有分支集都是 k-edge-connected,- 每个图 G 都是 O(qtloglogt)-colorable 或包含一个 Kt-minor 模型，其所有分支集都是 modulo-q-connected （即、通过考虑数图的循环超图，我们得到了关于具有大二色数的数图中强最小图的已知结果，并将其作为特例。我们还构造了二色数为 32(t-1) 的数图，这些数图不包含 t 个顶点上的完整数图作为强次要数，从而从反面回答了梅萨罗斯（Mészáros）和作者提出的问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

European Journal of Combinatorics 数学-数学

CiteScore

2.10

自引率

10.00%

发文量

124

审稿时长

4-8 weeks

期刊介绍： The European Journal of Combinatorics is a high standard, international, bimonthly journal of pure mathematics, specializing in theories arising from combinatorial problems. The journal is primarily open to papers dealing with mathematical structures within combinatorics and/or establishing direct links between combinatorics and other branches of mathematics and the theories of computing. The journal includes full-length research papers on important topics.