Asymptotics of local face distributions and the face distribution of the complete graph

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics Pub Date : 2024-06-27 DOI:10.1016/j.ejc.2024.104019

Jesse Campion Loth

{"title":"Asymptotics of local face distributions and the face distribution of the complete graph","authors":"Jesse Campion Loth","doi":"10.1016/j.ejc.2024.104019","DOIUrl":null,"url":null,"abstract":"<div><p>We are interested in the distribution of the number of faces across all the <span><math><mrow><mn>2</mn><mo>−</mo></mrow></math></span>cell embeddings of a graph, which is equivalent to the distribution of genus by Euler’s formula. In order to study this distribution, we consider the local distribution of faces at a single vertex. We show an asymptotic uniformity on this local face distribution which holds for any graph with large vertex degrees.</p><p>We use this to study the usual face distribution of the complete graph. We show that in this case, the local face distribution determines the face distribution for almost all of the whole graph. We use this result to show that a portion of the complete graph of size <span><math><mrow><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow><mrow><mo>|</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>|</mo></mrow></mrow></math></span> has the same face distribution as the set of all permutations, up to parity. Along the way, we prove new character bounds and an asymptotic uniformity on conjugacy class products.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824001045/pdfft?md5=fc1ded09367fd9c944d4c7c3dd000f19&pid=1-s2.0-S0195669824001045-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/S0195669824001045","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

We are interested in the distribution of the number of faces across all the $2 -$ cell embeddings of a graph, which is equivalent to the distribution of genus by Euler’s formula. In order to study this distribution, we consider the local distribution of faces at a single vertex. We show an asymptotic uniformity on this local face distribution which holds for any graph with large vertex degrees.

We use this to study the usual face distribution of the complete graph. We show that in this case, the local face distribution determines the face distribution for almost all of the whole graph. We use this result to show that a portion of the complete graph of size $(1 - o (1)) | K_{n} |$ has the same face distribution as the set of all permutations, up to parity. Along the way, we prove new character bounds and an asymptotic uniformity on conjugacy class products.

查看原文本刊更多论文

局部面分布的渐近性和完整图的面分布

我们感兴趣的是面的数量在图的所有 2 单元嵌入中的分布，根据欧拉公式，这等同于属的分布。为了研究这种分布，我们考虑了单个顶点上面的局部分布。我们展示了这种局部面分布的渐近均匀性，它对任何具有大顶点度的图都成立。我们证明，在这种情况下，局部面分布几乎决定了整个图的所有面分布。我们利用这一结果证明了大小为 (1-o(1))|Kn| 的完整图的一部分具有与所有排列集合相同的面分布，直至奇偶性。同时，我们还证明了共轭类积的新特征边界和渐近均匀性。

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.