On Packing Trees into Complete Bipartite Graphs

Q3 Multidisciplinary

Wuhan University Journal of Natural Sciences Pub Date : 2023-06-01 DOI:10.1051/wujns/2023283221

Yan Peng, Hong Wang

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引用次数: 0

Abstract

Let [see formula in PDF] denote the complete bipartite graph of order [see formula in PDF] with vertex partition sets [see formula in PDF] and [see formula in PDF]. We prove that for each tree [see formula in PDF] of order [see formula in PDF], there is a packing of [see formula in PDF] copies of [see formula in PDF] into a complete bipartite graph [see formula in PDF]. The ideal of the work comes from the "Tree packing conjecture" made by Gyráfás and Lehel. Bollobás confirmed the "Tree packing conjecture" for many small trees, who showed that one can pack [see formula in PDF] into [see formula in PDF] and that a better bound would follow from a famous conjecture of Erd[see formula in PDF]s. In a similar direction, Hobbs, Bourgeois and Kasiraj made the following conjecture: Any sequence of trees [see formula in PDF], … , [see formula in PDF], with [see formula in PDF] having order [see formula in PDF], can be packed into [see formula in PDF]. Further Hobbs, Bourgeois and Kasiraj proved that any two trees can be packed into a complete bipartite graph [see formula in PDF]. Motivated by these results, Wang Hong proposed the conjecture: For each tree [see formula in PDF] of order [see formula in PDF], there is a [see formula in PDF]⁃packing of [see formula in PDF] in some complete bipartite graph [see formula in PDF]. In this paper, we prove a weak version of this conjecture.

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关于将树包装成完全二部图的问题

令[见PDF公式]表示具有顶点划分集[见PDF公式]和[见PDF公式]的有序完全二部图[见PDF公式]。我们证明了对于每一个有序的树[见PDF中的公式]，有一个[见PDF中的公式]的副本的[见PDF中的公式]的包装成一个完全二部图[见PDF中的公式]。作品的理想来源于Gyráfás和Lehel的“树包装猜想”。Bollobás证实了许多小树的“树包装猜想”，他表明人们可以将[见PDF中的公式]打包成[见PDF中的公式]，并且从一个著名的Erd猜想(见PDF中的公式)中得出一个更好的边界。在类似的方向上，霍布斯，布尔乔亚和卡西拉提出了以下猜想:任何树的序列[见PDF中的公式]，…，[见PDF中的公式]，与[见PDF中的公式]有顺序[见PDF中的公式]，可以被打包成[见PDF中的公式]。霍布斯、布尔乔亚和卡西拉进一步证明了任意两棵树都可以被打包成一个完全二部图(见PDF中的公式)。基于这些结果，王宏提出了一个猜想:对于每一个有序的树[见公式]，在某完全二部图[见公式]中都存在一个⁃包装的[见公式]。在本文中，我们证明了这个猜想的一个弱版本。

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

Wuhan University Journal of Natural Sciences Multidisciplinary-Multidisciplinary

CiteScore

0.40

自引率

0.00%

发文量

2485

期刊介绍： Wuhan University Journal of Natural Sciences aims to promote rapid communication and exchange between the World and Wuhan University, as well as other Chinese universities and academic institutions. It mainly reflects the latest advances being made in many disciplines of scientific research in Chinese universities and academic institutions. The journal also publishes papers presented at conferences in China and abroad. The multi-disciplinary nature of Wuhan University Journal of Natural Sciences is apparent in the wide range of articles from leading Chinese scholars. This journal also aims to introduce Chinese academic achievements to the world community, by demonstrating the significance of Chinese scientific investigations.