Bipartite graphs with no K6 minor

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2023-09-20 DOI:10.1016/j.jctb.2023.08.005

Maria Chudnovsky , Alex Scott , Paul Seymour , Sophie Spirkl

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引用次数: 0

Abstract

A theorem of Mader shows that every graph with average degree at least eight has a $K_{6}$ minor, and this is false if we replace eight by any smaller constant. Replacing average degree by minimum degree seems to make little difference: we do not know whether all graphs with minimum degree at least seven have $K_{6}$ minors, but minimum degree six is certainly not enough. For every $ε > 0$ there are arbitrarily large graphs with average degree at least $8 - ε$ and minimum degree at least six, with no $K_{6}$ minor.

But what if we restrict ourselves to bipartite graphs? The first statement remains true: for every $ε > 0$ there are arbitrarily large bipartite graphs with average degree at least $8 - ε$ and no $K_{6}$ minor. But surprisingly, going to minimum degree now makes a significant difference. We will show that every bipartite graph with minimum degree at least six has a $K_{6}$ minor. Indeed, it is enough that every vertex in the larger part of the bipartition has degree at least six.

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无K6次的二部图

马德的一个定理表明，每个平均度至少为8的图都有一个K6次，如果我们用任何较小的常数代替8，这是错误的。用最小度代替平均度似乎没有什么区别：我们不知道是否所有最小度为7的图都有K6次，但最小度为6肯定是不够的。对于每个ε>；0有任意大的图，平均度至少为8-ε，最小度至少为6，没有K6次。但是，如果我们把自己限制在二分图上呢？第一种说法仍然成立：对于每个ε>；0存在任意大的二部图，其平均度至少为8-ε并且没有K6次。但令人惊讶的是，现在达到最低学历会产生显著的影响。我们将证明每一个最小度为6的二分图都有一个K6次图。事实上，在二分的较大部分中，每个顶点的度数至少为6就足够了。

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.