In this paper we show that every graph of pathwidth less than $k$ that has a path of order $n$ also has an induced path of order at least $\frac{1}{3} n^{1/k}$. This is an exponential improvement and a generalization of the polylogarithmic bounds obtained by Esperet, Lemoine and Maffray (2016) for interval graphs of bounded clique number. We complement this result with an upper-bound. This result is then used to prove the two following generalizations: - every graph of treewidth less than $k$ that has a path of order $n$ contains an induced path of order at least $\frac{1}{4} (\log n)^{1/k}$; - for every non-trivial graph class that is closed under topological minors there is a constant $d \in (0,1)$ such that every graph from this class that has a path of order $n$ contains an induced path of order at least $(\log n)^d$. We also describe consequences of these results beyond graph classes that are closed under topological minors.
翻译:在本文中,我们展示了每个路径小于1k$的路径图,该路径路径的顺序路径为$n美元。 这是 Esperet、 Lemoine 和 Maffray 获得的多面形框的指数性改进和概括。 我们用一个上限来补充这个结果。 这个结果被用来证明以下两种概括性: - 每棵树小于1k$的顺序路径为$n美元,其中含有至少$\frac{1QQQ$的顺序路径( log n) 1/k} 。 对于每个在表层下关闭的非三角形图类, 都有固定的 $ / 英寸 (0, 1美元), 这样每个有顺序路径的图表都含有至少$( log n) ⁇ d$的顺序路径。 我们还描述了这些结果在表层下关闭的未成年人的结果。