Improved Bounds for Weak Coloring Numbers

Weak coloring numbers generalize the notion of degeneracy a graph. They were introduced by Kierstead & Yang in context games on graphs. Recently, several connections have been uncovered between weak and various parameters studied graph minor theory its generalizations. In this note, we show that for every fixed $k\geq1$, maximum $r$-th number with simple treewidth $k$ is $\Theta(r^{k-1}\log r)$. As corollary, improve lower bound planar graphs from $\Omega(r^2)$ to $\Omega(r^2\log r)$, obtain tight $\Theta(r\log r)$ outerplanar