TY - GEN

T1 - Weak Coloring Numbers of Intersection Graphs

AU - Dvorák, Zdenek

AU - Pekárek, Jakub

AU - Ueckerdt, Torsten

AU - Yuditsky, Yelena

N1 - Funding Information:
Funding Zdeněk Dvořák: Supported by the ERC-CZ project LL2005 (Algorithms and complexity within and beyond bounded expansion) of the Ministry of Education of Czech Republic. Jakub Pekárek: Supported by the ERC-CZ project LL2005 (Algorithms and complexity within and beyond bounded expansion) of the Ministry of Education of Czech Republic.
Publisher Copyright:
© Zdenk Dvok, Jakub Pekrek, Torsten Ueckerdt, and Yelena Yuditsky; licensed under Creative Commons License CC-BY 4.0

PY - 2022/6/1

Y1 - 2022/6/1

N2 - 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, Dvorák, McCarty, and Norin observed a natural volume-based upper bound for the strong coloring numbers of intersection graphs of well-behaved objects in Rd, 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).

AB - 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, Dvorák, McCarty, and Norin observed a natural volume-based upper bound for the strong coloring numbers of intersection graphs of well-behaved objects in Rd, 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).

KW - geometric intersection graphs

KW - strong coloring numbers

KW - weak

UR - http://www.scopus.com/inward/record.url?scp=85134323708&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.SoCG.2022.39

DO - 10.4230/LIPIcs.SoCG.2022.39

M3 - Conference contribution

AN - SCOPUS:85134323708

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 38th International Symposium on Computational Geometry, SoCG 2022

A2 - Goaoc, Xavier

A2 - Kerber, Michael

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 38th International Symposium on Computational Geometry, SoCG 2022

Y2 - 7 June 2022 through 10 June 2022

ER -