License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2022.39
URN: urn:nbn:de:0030-drops-160477
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16047/
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Dvořák, Zdeněk ; Pekárek, Jakub ; Ueckerdt, Torsten ; Yuditsky, Yelena

Weak Coloring Numbers of Intersection Graphs

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LIPIcs-SoCG-2022-39.pdf (0.9 MB)

Abstract

Weak and strong coloring numbers are generalizations of the degeneracy of a graph, where for a positive integer k, we seek a vertex ordering such that every vertex can (weakly respectively strongly) reach in k steps only few vertices that precede it in the ordering. Both notions capture the sparsity of a graph or a graph class, and have interesting applications in structural and algorithmic graph theory. Recently, Dvořák, McCarty, and Norin observed a natural volume-based upper bound for the strong coloring numbers of intersection graphs of well-behaved objects in ℝ^d, such as homothets of a compact convex object, or comparable axis-aligned boxes.
In this paper, we prove upper and lower bounds for the k-th weak coloring numbers of these classes of intersection graphs. As a consequence, we describe a natural graph class whose strong coloring numbers are polynomial in k, but the weak coloring numbers are exponential. We also observe a surprising difference in terms of the dependence of the weak coloring numbers on the dimension between touching graphs of balls (single-exponential) and hypercubes (double-exponential).

BibTeX - Entry

@InProceedings{dvorak_et_al:LIPIcs.SoCG.2022.39,
  author =	{Dvo\v{r}\'{a}k, Zden\v{e}k and Pek\'{a}rek, Jakub and Ueckerdt, Torsten and Yuditsky, Yelena},
  title =	{{Weak Coloring Numbers of Intersection Graphs}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{39:1--39:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16047},
  URN =		{urn:nbn:de:0030-drops-160477},
  doi =		{10.4230/LIPIcs.SoCG.2022.39},
  annote =	{Keywords: geometric intersection graphs, weak and strong coloring numbers}
}

Keywords: geometric intersection graphs, weak and strong coloring numbers
Collection: 38th International Symposium on Computational Geometry (SoCG 2022)
Issue Date: 2022
Date of publication: 01.06.2022

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