Disjointness graphs of short polygonal chains

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2023-09-18 DOI:10.1016/j.jctb.2023.08.008

János Pach , Gábor Tardos , Géza Tóth

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引用次数: 0

Abstract

The disjointness graph of a set system is a graph whose vertices are the sets, two being connected by an edge if and only if they are disjoint. It is known that the disjointness graph G of any system of segments in the plane is χ-bounded, that is, its chromatic number $χ (G)$ is upper bounded by a function of its clique number $ω (G)$ .

Here we show that this statement does not remain true for systems of polygonal chains of length 2. We also construct systems of polygonal chains of length 3 such that their disjointness graphs have arbitrarily large girth and chromatic number. In the opposite direction, we show that the class of disjointness graphs of (possibly self-intersecting) 2-way infinite polygonal chains of length 3 is χ-bounded: for every such graph G, we have $χ (G) \leq {(ω (G))}^{3} + ω (G)$ .

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多边形短链的不连续图

集合系统的不相交图是一个顶点是集合的图，两个顶点通过边连接，当且仅当它们不相交。已知平面上任意一个分段系统的不相交图G是χ-有界的，即其色数χ（G）是其团数ω（G）的函数的上界。我们还构造了长度为3的多边形链的系统，使得它们的不相交图具有任意大的周长和色数。在相反的方向上，我们证明了一类长度为3的（可能自相交的）双向无限多边形链的不相交图是χ-有界的：对于每一个这样的图G，我们都有χ（G）≤（ω（G））3+Ω（G）。

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.