Graphs of bounded degeneracy are known to contain induced paths of order $\Omega(\log \log n)$ when they contain a path of order $n$, as proved by Ne\v{s}et\v{r}il and Ossona de Mendez (2012). In 2016 Esperet, Lemoine, and Maffray conjectured that this bound could be improved to $\Omega((\log n)^c)$ 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.

翻译：有界退化度的图被证明包含阶为 $\Omega(\log \log n)$ 的诱导路径（当它们包含一条阶为 $n$ 的路径时），该结果由 Nešetřil 和 Ossona de Mendez (2012) 证明。2016年，Esperet、Lemoine 和 Maffray 猜想这一下界可改进为 $\Omega((\log n)^c)$，其中常数 $c>0$ 依赖于退化度。我们通过构造任意大值 $n$ 的图反驳了这一猜想：该图为 2-退化图，包含一条阶为 $n$ 的路径，但所有诱导路径的阶均为 $O((\log \log n)^2)$。我们还证明所构造的图具有线性有界的染色数。