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Colouring and Covering Nowhere Dense Graphs

Part of the Lecture Notes in Computer Science book series (LNTCS,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 \(r\ge 3\).

Keywords

  • Colouring Numbers
  • Bounded Tree Width
  • Density Classes
  • Graph Classes
  • Treewidth

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Correspondence to Konstantinos Stavropoulos .

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Grohe, M., Kreutzer, S., Rabinovich, R., Siebertz, S., Stavropoulos, K. (2016). Colouring and Covering Nowhere Dense Graphs. In: Mayr, E. (eds) Graph-Theoretic Concepts in Computer Science. WG 2015. Lecture Notes in Computer Science(), vol 9224. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53174-7_23

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  • DOI: https://doi.org/10.1007/978-3-662-53174-7_23

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