We study the growth rate of weak coloring numbers of graphs excluding a fixed graph as a minor. Van den Heuvel et al. (European J. of Combinatorics, 2017) showed that for a fixed graph $X$, the maximum $r$-th weak coloring number of $X$-minor-free graphs is polynomial in $r$. We determine this polynomial up to a factor of $\mathcal{O}(r \log r)$. Moreover, we tie the exponent of the polynomial to a structural property of $X$, namely, $2$-treedepth. As a result, for a fixed graph $X$ and an $X$-minor-free graph $G$, we show that $\mathrm{wcol}_r(G)= \mathcal{O}(r^{\mathrm{td}(X)-1}\mathrm{log}\ r)$, which improves on the bound $\mathrm{wcol}_r(G) = \mathcal{O}(r^{g(\mathrm{td}(X))})$ given by Dujmovi\'c et al. (SODA, 2024), where $g$ is an exponential function. In the case of planar graphs of bounded treewidth, we show that the maximum $r$-th weak coloring number is in $\mathcal{O}(r^2\mathrm{log}\ r$), which is best possible.

翻译：我们研究了排除固定图作为子式的图的弱染色数的增长率。Van den Heuvel等人（European J. of Combinatorics, 2017）证明了对于固定图$X$，排除$X$子式的图的最大第$r$弱染色数是$r$的多项式。我们将该多项式确定到$\mathcal{O}(r \log r)$因子内。此外，我们将该多项式的指数与$X$的一个结构性质，即$2$-树深，联系起来。因此，对于固定图$X$和排除$X$子式的图$G$，我们证明了$\mathrm{wcol}_r(G)= \mathcal{O}(r^{\mathrm{td}(X)-1}\mathrm{log}\ r)$，这改进了Dujmovi\'c等人（SODA, 2024）给出的界$\mathrm{wcol}_r(G) = \mathcal{O}(r^{g(\mathrm{td}(X))})$，其中$g$是一个指数函数。对于有界树宽的平面图，我们证明了最大第$r$弱染色数属于$\mathcal{O}(r^2\mathrm{log}\ r$)，这是最优的。