Bounding generalized coloring numbers of planar graphs using coin models

TL;DR

This paper investigates Koebe orderings of planar graphs modeled via coin representations, establishing bounds on their generalized coloring numbers that match known lower bounds, thus advancing understanding of planar graph colorability.

Contribution

It provides new bounds on the $d$-admissibility and weak $d$-coloring numbers of planar graphs using coin models, matching known lower bounds.

Findings

Bound $d$-admissibility by $O(d/ ext{ln} d)$

Bound weak $d$-coloring number by $O(d^4 ext{ln} d)$

Matches asymptotic lower bounds for planar graphs

Abstract

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 $d\in \mathbb{N}$, 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 Dvo\v{r}\'ak and Siebertz.