Weak coloring numbers of minor-closed graph classes

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 Dujmović 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.