Sparse graphs without long induced paths

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-01-05 DOI:10.1016/j.jctb.2023.12.003

Oscar Defrain , Jean-Florent Raymond

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引用次数: 0

Abstract

Graphs of bounded degeneracy are known to contain induced paths of order $Ω (\log \log n)$ when they contain a path of order n, as proved by Nešetřil and Ossona de Mendez (2012). In 2016 Esperet, Lemoine, and Maffray conjectured that this bound could be improved to $Ω ({(\log n)}^{c})$ for some constant $c > 0$ depending on the degeneracy.

We disprove this conjecture by constructing, for arbitrarily large values of n, a graph that is 2-degenerate, has a path of order n, and where all induced paths have order $O ({(\log \log n)}^{2})$ . We also show that the graphs we construct have linearly bounded coloring numbers.

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没有长诱导路径的稀疏图

已知有界退化图在包含阶数为 n 的路径时，会包含阶数为Ω(loglogn)的诱导路径，Nešetřil 和 Ossona de Mendez（2012 年）证明了这一点。2016年，Esperet、Lemoine和Maffray猜想，对于某个常数c>0（取决于退化程度），这个约束可以改进为Ω((logn)c)。我们推翻了这个猜想，为任意大的n值构造了一个图，它是2退化的，有一条阶数为n的路径，并且所有诱导路径的阶数都是O((loglogn)2)。我们还证明了我们构建的图具有线性有界着色数。

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.