An oriented discrepancy version of Dirac's theorem

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-07-22 DOI:10.1016/j.jctb.2024.06.008

Andrea Freschi, Allan Lo

{"title":"An oriented discrepancy version of Dirac's theorem","authors":"Andrea Freschi, Allan Lo","doi":"10.1016/j.jctb.2024.06.008","DOIUrl":null,"url":null,"abstract":"<div>The study of graph discrepancy problems, initiated by Erdős in the 1960s, has received renewed attention in recent years. In general, given a 2-edge-coloured graph G, one is interested in embedding a copy of a graph H in G with large discrepancy (i.e. the copy of H contains significantly more than half of its edges in one colour).Motivated by this line of research, Gishboliner, Krivelevich and Michaeli considered an oriented version of graph discrepancy for Hamilton cycles. In particular, they conjectured the following generalisation of Dirac's theorem: if G is an oriented graph on <math><mi>n</mi><mo>≥</mo><mn>3</mn></math> vertices with <math><mi>δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mi>n</mi><mo>/</mo><mn>2</mn></math>, then G contains a Hamilton cycle with at least <math><mi>δ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math> edges pointing forwards. In this paper, we present a full resolution to this conjecture.</div>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0095895624000571/pdfft?md5=c4578c27cd9e214edf177c194b1972bb&pid=1-s2.0-S0095895624000571-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0095895624000571","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}

引用次数: 0

Abstract

The study of graph discrepancy problems, initiated by Erdős in the 1960s, has received renewed attention in recent years. In general, given a 2-edge-coloured graph G, one is interested in embedding a copy of a graph H in G with large discrepancy (i.e. the copy of H contains significantly more than half of its edges in one colour).

Motivated by this line of research, Gishboliner, Krivelevich and Michaeli considered an oriented version of graph discrepancy for Hamilton cycles. In particular, they conjectured the following generalisation of Dirac's theorem: if G is an oriented graph on $n \geq 3$ vertices with $δ (G) \geq n / 2$ , then G contains a Hamilton cycle with at least $δ (G)$ edges pointing forwards. In this paper, we present a full resolution to this conjecture.

查看原文本刊更多论文

狄拉克定理的定向差异版本

图差异问题的研究由 Erdős 在 20 世纪 60 年代发起，近年来再次受到关注。一般来说，给定一个两边着色的图 G，人们感兴趣的是在 G 中嵌入一个具有较大差异的图 H 副本（即 H 副本含有明显超过一半的边为一种颜色）。特别是，他们猜想了狄拉克定理的以下概括：如果 G 是 n≥3 个顶点上的δ(G)≥n/2 的定向图，那么 G 包含一个至少有 δ(G) 条边指向前方的汉密尔顿循环。在本文中，我们提出了这一猜想的完整解决方案。

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

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.