稀疏图形多副本的拉姆齐数

IF 0.9 3区 数学 Q2 MATHEMATICS
Aurelio Sulser, Miloš Trujić
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引用次数: 0

摘要

对于一个图和一个整数 ,我们用 表示其副本的不相交联合。 1975 年,伯尔、厄多斯和斯宾塞开始研究拉姆齐数,拉姆齐数是目前已知拉姆齐数的少数实例之一。他们证明,只要拉姆齐数足够大,就会有一个常数使得 , 。随后,伯尔给出了一种隐含的计算方法,并指出这种长期行为发生在......的三倍指数时。最近,布契奇和苏达科夫重新提出了这个问题,并建立了一个基本严密的约束,表明当副本数仅为单指数时,这种行为已经出现。在稀疏图的情况下,我们提供了明显更强的约束,最明显的是有界最大度。这些约束与当前最先进的约束是相关的,而且(在某种程度上)是紧密的。我们的方法依赖于 Graham、Rödl 和 Ruciński 的一个漂亮的经典证明,重点是为有界度图开发一种高效的吸收方法。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

Ramsey numbers for multiple copies of sparse graphs

Ramsey numbers for multiple copies of sparse graphs

For a graph H and an integer n , we let n H denote the disjoint union of n copies of H . In 1975, Burr, Erdős and Spencer initiated the study of Ramsey numbers for n H , one of few instances for which Ramsey numbers are now known precisely. They showed that there is a constant c = c ( H ) such that r ( n H ) = ( 2 H α ( H ) ) n + c , provided n is sufficiently large. Subsequently, Burr gave an implicit way of computing c and noted that this long-term behaviour occurs when n is triply exponential in H . Very recently, Bucić and Sudakov revived the problem and established an essentially tight bound on n by showing r ( n H ) follows this behaviour already when the number of copies is just a single exponential. We provide significantly stronger bounds on n in case H is a sparse graph, most notably of bounded maximum degree. These are relatable to the current state-of-the-art bounds on r ( H ) and (in a way) tight. Our methods rely on a beautiful classic proof of Graham, Rödl and Ruciński, with an emphasis on developing an efficient absorbing method for bounded degree graphs.

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来源期刊
Journal of Graph Theory
Journal of Graph Theory 数学-数学
CiteScore
1.60
自引率
22.20%
发文量
130
审稿时长
6-12 weeks
期刊介绍: The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .
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