The Erdős-Gyárfás function f(n,4,5)=56n+o(n) — So Gyárfás was right

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B Pub Date : 2024-07-22 DOI:10.1016/j.jctb.2024.07.001

Patrick Bennett , Ryan Cushman , Andrzej Dudek , Paweł Prałat

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引用次数: 0

Abstract

A $(4, 5)$ -coloring of $K_{n}$ is an edge-coloring of $K_{n}$ where every 4-clique spans at least five colors. We show that there exist $(4, 5)$ -colorings of $K_{n}$ using $\frac{5}{6} n + o (n)$ colors. This settles a disagreement between Erdős and Gyárfás reported in their 1997 paper. Our construction uses a randomized process which we analyze using the so-called differential equation method to establish dynamic concentration. In particular, our coloring process uses random triangle removal, a process first introduced by Bollobás and Erdős, and analyzed by Bohman, Frieze and Lubetzky.

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Erdős-Gyárfás 函数 f(n,4,5)=56n+o(n) - 所以 Gyárfás 是对的

Kn 的 (4,5)- 着色是 Kn 的边染色，其中每个 4 小格至少跨越五种颜色。我们证明存在使用 56n+o(n) 种颜色的 Kn (4,5)- 着色。这解决了 Erdős 和 Gyárfás 在 1997 年论文中提出的分歧。我们的构造使用了一种随机过程，我们使用所谓的微分方程法对其进行分析，以建立动态集中。特别是，我们的着色过程使用了随机三角形去除，这个过程最早由 Bollobás 和 Erdős 提出，并由 Bohman、Frieze 和 Lubetzky 进行了分析。

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来源期刊

Journal of Combinatorial Theory Series B 数学-数学

CiteScore

2.70

自引率

14.30%

发文量

审稿时长

6-12 weeks

期刊介绍： The Journal of Combinatorial Theory publishes original mathematical research dealing with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series B is concerned primarily with graph theory and matroid theory and is a valuable tool for mathematicians and computer scientists.