Sets of r-Graphs that Color All r-Graphs

IF 1 2区 数学 Q1 MATHEMATICS
Yulai Ma, Davide Mattiolo, Eckhard Steffen, Isaak H. Wolf
{"title":"Sets of r-Graphs that Color All r-Graphs","authors":"Yulai Ma, Davide Mattiolo, Eckhard Steffen, Isaak H. Wolf","doi":"10.1007/s00493-025-00144-4","DOIUrl":null,"url":null,"abstract":"<p>An <i>r</i>-regular graph is an <i>r</i>-graph, if every odd set of vertices is connected to its complement by at least <i>r</i> edges. Let <i>G</i> and <i>H</i> be <i>r</i>-graphs. An <i>H</i><i>-coloring</i> of <i>G</i> is a mapping <span>\\(f:E(G) \\rightarrow E(H)\\)</span> such that each <i>r</i> adjacent edges of <i>G</i> are mapped to <i>r</i> adjacent edges of <i>H</i>. For every <span>\\(r\\ge 3\\)</span>, let <span>\\(\\mathcal H_r\\)</span> be an inclusion-wise minimal set of connected <i>r</i>-graphs, such that for every connected <i>r</i>-graph <i>G</i> there is an <span>\\(H \\in \\mathcal H_r\\)</span> which colors <i>G</i>. The Petersen Coloring Conjecture states that <span>\\(\\mathcal H_3\\)</span> consists of the Petersen graph <i>P</i>. We show that if true, then this is a very exclusive situation. Our main result is that either <span>\\(\\mathcal H_3 = \\{P\\}\\)</span> or <span>\\(\\mathcal H_3\\)</span> is an infinite set and if <span>\\(r \\ge 4\\)</span>, then <span>\\(\\mathcal H_r\\)</span> is an infinite set. In particular, for all <span>\\(r \\ge 3\\)</span>, <span>\\(\\mathcal H_r\\)</span> is unique. We first characterize <span>\\(\\mathcal H_r\\)</span> and then prove that if <span>\\(\\mathcal H_r\\)</span> contains more than one element, then it is an infinite set. To obtain our main result we show that <span>\\(\\mathcal H_r\\)</span> contains the smallest <i>r</i>-graphs of class 2 and the smallest poorly matchable <i>r</i>-graphs, and we determine the smallest <i>r</i>-graphs of class 2.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"213 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Combinatorica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00493-025-00144-4","RegionNum":2,"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

An r-regular graph is an r-graph, if every odd set of vertices is connected to its complement by at least r edges. Let G and H be r-graphs. An H-coloring of G is a mapping \(f:E(G) \rightarrow E(H)\) such that each r adjacent edges of G are mapped to r adjacent edges of H. For every \(r\ge 3\), let \(\mathcal H_r\) be an inclusion-wise minimal set of connected r-graphs, such that for every connected r-graph G there is an \(H \in \mathcal H_r\) which colors G. The Petersen Coloring Conjecture states that \(\mathcal H_3\) consists of the Petersen graph P. We show that if true, then this is a very exclusive situation. Our main result is that either \(\mathcal H_3 = \{P\}\) or \(\mathcal H_3\) is an infinite set and if \(r \ge 4\), then \(\mathcal H_r\) is an infinite set. In particular, for all \(r \ge 3\), \(\mathcal H_r\) is unique. We first characterize \(\mathcal H_r\) and then prove that if \(\mathcal H_r\) contains more than one element, then it is an infinite set. To obtain our main result we show that \(\mathcal H_r\) contains the smallest r-graphs of class 2 and the smallest poorly matchable r-graphs, and we determine the smallest r-graphs of class 2.

为所有r-图着色的r-图集合
一个r正则图是一个r图,如果每一个奇数顶点集与它的补集至少有r条边相连。设G和H是r图。G的h染色是一个映射\(f:E(G) \rightarrow E(H)\),使得G的每r个相邻边都映射到h的r个相邻边。对于每一个\(r\ge 3\),设\(\mathcal H_r\)是连通r图的包含最小集,使得对于每一个连通r图G都有一个\(H \in \mathcal H_r\)为G着色。Petersen着色猜想表明\(\mathcal H_3\)由Petersen图p组成,我们证明如果成立,那么这是一个非常排斥的情况。我们的主要结果是\(\mathcal H_3 = \{P\}\)或\(\mathcal H_3\)是一个无限集,如果\(r \ge 4\),则\(\mathcal H_r\)是一个无限集。特别是,对于所有\(r \ge 3\), \(\mathcal H_r\)都是独一无二的。我们首先刻画\(\mathcal H_r\),然后证明如果\(\mathcal H_r\)包含多于一个元素,那么它是一个无限集。为了得到我们的主要结果,我们证明\(\mathcal H_r\)包含2类最小的r-图和最小的差匹配r-图,并确定了2类最小的r-图。
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
Combinatorica
Combinatorica 数学-数学
CiteScore
1.90
自引率
0.00%
发文量
45
审稿时长
>12 weeks
期刊介绍: COMBINATORICA publishes research papers in English in a variety of areas of combinatorics and the theory of computing, with particular emphasis on general techniques and unifying principles. Typical but not exclusive topics covered by COMBINATORICA are - Combinatorial structures (graphs, hypergraphs, matroids, designs, permutation groups). - Combinatorial optimization. - Combinatorial aspects of geometry and number theory. - Algorithms in combinatorics and related fields. - Computational complexity theory. - Randomization and explicit construction in combinatorics and algorithms.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信