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Hyperbolic Random Graphs: Clique Number and Degeneracy with Implications for Colouring

Authors Samuel Baguley , Yannic Maus , Janosch Ruff , George Skretas

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Author Details

Samuel Baguley
  • Hasso Plattner Institute, University of Potsdam, Germany
Yannic Maus
  • TU Graz, Austria
Janosch Ruff
  • Hasso Plattner Institute, University of Potsdam, Germany
George Skretas
  • Hasso Plattner Institute, University of Potsdam, Germany

Funding

This research was partially funded by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) – project number 390859508, and by Austrian Science Fund (FWF) https://doi.org/10.55776/P36280, https://doi.org/10.55776/I6915, and https://doi.org/10.55776/DOC183.

Acknowledgements

The authors would like to thank Thomas Bläsius for enriching discussions and Maximilian Katzmann for running experiments indicating a gap between core and maximum inner-neighbourhood during a research visit of JR in Karlsruhe.

Cite As Get BibTex

Samuel Baguley, Yannic Maus, Janosch Ruff, and George Skretas. Hyperbolic Random Graphs: Clique Number and Degeneracy with Implications for Colouring. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 13:1-13:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.13

Abstract

Hyperbolic random graphs inherit many properties that are present in real-world networks. The hyperbolic geometry imposes a scale-free network with a strong clustering coefficient. Other properties like a giant component, the small world phenomena and others follow. This motivates the design of simple algorithms for hyperbolic random graphs. 
In this paper we consider threshold hyperbolic random graphs (HRGs). Greedy heuristics are commonly used in practice as they deliver a good approximations to the optimal solution even though their theoretical analysis would suggest otherwise. A typical example for HRGs are degeneracy-based greedy algorithms [Bläsius, Fischbeck; Transactions of Algorithms '24]. In an attempt to bridge this theory-practice gap we characterise the parameter of degeneracy yielding a simple approximation algorithm for colouring HRGs. The approximation ratio of our algorithm ranges from (2/√3) to 4/3 depending on the power-law exponent of the model. We complement our findings for the degeneracy with new insights on the clique number of hyperbolic random graphs. We show that degeneracy and clique number are substantially different and derive an improved upper bound on the clique number. Additionally, we show that the core of HRGs does not constitute the largest clique. 
Lastly we demonstrate that the degeneracy of the closely related standard model of geometric inhomogeneous random graphs behaves inherently different compared to the one of hyperbolic random graphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Random network models
Keywords
  • hyperbolic random graphs
  • scale-free networks
  • power-law graphs
  • cliques
  • degeneracy
  • vertex colouring
  • chromatic number

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References

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