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Colouring and Covering Nowhere Dense Graphs

  • Martin Grohe
  • Stephan Kreutzer
  • Roman Rabinovich
  • Sebastian Siebertz
  • Konstantinos Stavropoulos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9224)

Abstract

In [9] it was shown that nowhere dense classes of graphs admit sparse neighbourhood covers of small degree. We show that a monotone graph class admits sparse neighbourhood covers if and only if it is nowhere dense. The existence of such covers for nowhere dense classes is established through bounds on so-called weak colouring numbers.

The core results of this paper are various lower and upper bounds on the weak colouring numbers and other, closely related generalised colouring numbers. We prove tight bounds for these numbers on graphs of bounded tree width. We clarify and tighten the relation between the expansion (in the sense of “bounded expansion” [15]) and the various generalised colouring numbers. These upper bounds are complemented by new, stronger exponential lower bounds on the generalised colouring numbers. Finally, we show that computing weak r-colouring numbers is NP-complete for all \(r\ge 3\).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Martin Grohe
    • 1
  • Stephan Kreutzer
    • 2
  • Roman Rabinovich
    • 2
  • Sebastian Siebertz
    • 2
  • Konstantinos Stavropoulos
    • 1
  1. 1.RWTH Aachen UniversityAachenGermany
  2. 2.Technical University BerlinBerlinGermany

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