Graphs of bounded degeneracy are known to contain induced paths of order $\mathrm{\Omega }(\log \log n)$ 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 $\mathrm{\Omega }({(\log n)}^c)$ 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.

中文翻译：

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没有长诱导路径的稀疏图

已知有界简并图包含有序诱导路径$\mathrm{\Omega }（\mathrm{日志}\mathrm{日志}n）$当它们包含n阶路径时，正如 Nešetřil 和 Ossona de Mendez (2012) 所证明的那样。2016 年，Esperet、Lemoine 和 Maffray 推测这个界限可以改进为$\mathrm{\Omega }（{（\mathrm{日志}n）}^C）$对于一些常数$C>0$取决于退化程度。

我们通过对于任意大的n值构造一个 2-简并图来反驳这个猜想，该图具有阶数为n的路径，并且所有诱导路径都具有阶数$氧（{（\mathrm{日志}\mathrm{日志}n）}^2）$。我们还表明，我们构建的图具有线性有界的着色数。