On locally rainbow colourings

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-06-21 DOI:10.1016/j.jctb.2024.06.003

Barnabás Janzer , Oliver Janzer

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引用次数: 0

Abstract

Given a graph H, let $g (n, H)$ denote the smallest k for which the following holds. We can assign a k-colouring $f_{v}$ of the edge set of $K_{n}$ to each vertex v in $K_{n}$ with the property that for any copy T of H in $K_{n}$ , there is some $u \in V (T)$ such that every edge in T has a different colour in $f_{u}$ .

The study of this function was initiated by Alon and Ben-Eliezer. They characterized the family of graphs H for which $g (n, H)$ is bounded and asked whether it is true that for every other graph $g (n, H)$ is polynomial. We show that this is not the case and characterize the family of connected graphs H for which $g (n, H)$ grows polynomially. Answering another question of theirs, we also prove that for every $ε > 0$ , there is some $r = r (ε)$ such that $g (n, K_{r}) \geq n^{1 - ε}$ for all sufficiently large n.

Finally, we show that the above problem is connected to the Erdős–Gyárfás function in Ramsey Theory, and prove a family of special cases of a conjecture of Conlon, Fox, Lee and Sudakov by showing that for each fixed r the complete r-uniform hypergraph $K_{n}^{(r)}$ can be edge-coloured using a subpolynomial number of colours in such a way that at least r colours appear among any $r + 1$ vertices.

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关于局部彩虹色

给定一个图 H，让 g(n,H) 表示以下条件成立的最小 k。我们可以为 Kn 中的每个顶点 v 指定 Kn 边集的 k 颜色 fv，其性质是：对于 Kn 中 H 的任意副本 T，存在某个 u∈V(T)，使得 T 中的每条边在 fu 中都有不同的颜色。他们描述了 g(n,H) 是有界的图 H 族的特征，并询问对于其他所有图，g(n,H) 是否都是多项式。我们证明情况并非如此，并描述了 g(n,H) 多项式增长的连通图 H 族的特征。为了回答他们的另一个问题，我们还证明了对于每一个 ε>0，存在某个 r=r(ε)，使得对于所有足够大的 n，g(n,Kr)≥n1-ε。最后，我们证明了上述问题与拉姆齐理论中的厄尔多斯-吉亚法函数相关联，并证明了康伦、福克斯、李和苏达科夫猜想的一系列特例，即对于每个固定的 r，完整的 r-Uniform 超图 Kn(r) 可以用亚对数个颜色进行边着色，从而在任意 r+1 个顶点中至少出现 r 个颜色。

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

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.