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arXiv · 2201.09340

Bounding generalized coloring numbers of planar graphs using coin models

Jesper Nederlof·Michał Pilipczuk·Karol Węgrzycki
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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.

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Jesper Nederlof, Michał Pilipczuk, Karol Węgrzycki. 2022-01-23. Bounding generalized coloring numbers of planar graphs using coin models. https://arxiv.org/abs/2201.09340

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