没有长诱导路径的稀疏图

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B 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

摘要

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。

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Sparse graphs without long induced paths

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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来源期刊

Journal of Combinatorial Theory Series B 数学-数学

CiteScore

2.70

自引率

14.30%

发文量

审稿时长

6-12 weeks

期刊介绍： The Journal of Combinatorial Theory publishes original mathematical research dealing with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series B is concerned primarily with graph theory and matroid theory and is a valuable tool for mathematicians and computer scientists.