On a rainbow extremal problem for color‐critical graphs

IF 0.9 3区数学 Q4 COMPUTER SCIENCE, SOFTWARE ENGINEERING

Random Structures & Algorithms Pub Date : 2023-10-17 DOI:10.1002/rsa.21189

Debsoumya Chakraborti, Jaehoon Kim, Hyunwoo Lee, Hong Liu, Jaehyeon Seo

引用次数: 8

Abstract

Abstract Given graphs over a common vertex set of size , what is the maximum value of having no “colorful” copy of , that is, a copy of containing at most one edge from each ? Keevash, Saks, Sudakov, and Verstraëte denoted this number as and completely determined for large . In fact, they showed that, depending on the value of , one of the two natural constructions is always the extremal construction. Moreover, they conjectured that the same holds for every color‐critical graphs, and proved it for 3‐color‐critical graphs. They also asked to classify the graphs that have only these two extremal constructions. We prove their conjecture for 4‐color‐critical graphs and for almost all ‐color‐critical graphs when . Moreover, we show that for every non‐color‐critical non‐bipartite graphs, none of the two natural constructions is extremal for certain values of .

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色临界图的彩虹极值问题

给定一个大小为公共顶点集的图，没有“彩色”副本的最大值是多少，也就是说，每个副本最多包含一条边?Keevash, Saks, Sudakov和Verstraëte表示这个数字为，并且完全确定为大。事实上，他们表明，根据值的不同，两种自然结构中的一种总是极端结构。此外，他们推测同样的定理适用于所有色临界图，并证明了它适用于3色临界图。他们还要求对只有这两个极值结构的图进行分类。我们在4色临界图和几乎所有色临界图上证明了他们的猜想。此外，我们证明了对于每一个非色临界非二部图，对于的某些值，两种自然结构都不是极值的。

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

Random Structures & Algorithms 数学-计算机：软件工程

CiteScore

2.50

自引率

10.00%

发文量

审稿时长

>12 weeks

期刊介绍： It is the aim of this journal to meet two main objectives: to cover the latest research on discrete random structures, and to present applications of such research to problems in combinatorics and computer science. The goal is to provide a natural home for a significant body of current research, and a useful forum for ideas on future studies in randomness. Results concerning random graphs, hypergraphs, matroids, trees, mappings, permutations, matrices, sets and orders, as well as stochastic graph processes and networks are presented with particular emphasis on the use of probabilistic methods in combinatorics as developed by Paul Erdõs. The journal focuses on probabilistic algorithms, average case analysis of deterministic algorithms, and applications of probabilistic methods to cryptography, data structures, searching and sorting. The journal also devotes space to such areas of probability theory as percolation, random walks and combinatorial aspects of probability.