Bounding Generalized Coloring Numbers of Planar Graphs Using Coin Models

Abstract

We study Koebe orderings of planar graphs: vertex orderings obtained bymodelling the graph as the intersection graph of pairwise internally-disjointdiscs in the plane, and ordering the vertices by non-increasing radii of theassociated discs. We prove that for every $d\in \mathbb{N}$, any such orderinghas $d$-admissibility bounded by $O(d/\ln d)$ and weak $d$-coloring numberbounded by $O(d^4 \ln d)$. This in particular shows that the $d$-admissibilityof planar graphs is bounded by $O(d/\ln d)$, which asymptotically matches aknown lower bound due to Dvo\v{r}\'ak and Siebertz.<br