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.
Categories: math.CO cs.DM