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 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 wcolr(G ) = O (rtd(X )-1 log r ), which improves on the bound wcolr(G ) = O (rg(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 O (r2 log r ), which is best possible.