Bounding Generalized Coloring Numbers of Planar Graphs Using Coin Models

We study Koebe orderings of planar graphs: vertex orderings obtained by modelling the graph as the intersection graph of pairwise internally-disjoint discs in the plane, and ordering the vertices by non-increasing radii of the associated discs. We prove that for every d is an element of N, any such ordering has d-admissibility bounded by O(d/ ln d) and weak d-coloring number bounded by O(d(4)ln d). This in particular shows that the d-admissibility of planar graphs is bounded by O(d/ ln d), which asymptotically matches a known lower bound due to Dvorak and Siebertz.