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Canonical Labeling of Sparse Random Graphs

Authors Oleg Verbitsky , Maksim Zhukovskii

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ACM Subject Classification
  • Mathematics of computing → Random graphs
  • Mathematics of computing → Graph algorithms
Keywords
  • Graph isomorphism
  • random graphs
  • canonical labeling
  • color refinement

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Abstract

We show that if p = O(1/n), then the Erdős-Rényi random graph G(n,p) with high probability admits a canonical labeling computable in time O(nlog n). Combined with the previous results on the canonization of random graphs, this implies that G(n,p) with high probability admits a polynomial-time canonical labeling whatever the edge probability function p. Our algorithm combines the standard color refinement routine with simple post-processing based on the classical linear-time tree canonization. Noteworthy, our analysis of how well color refinement performs in this setting allows us to complete the description of the automorphism group of the 2-core of G(n,p).

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Oleg Verbitsky and Maksim Zhukovskii. Canonical Labeling of Sparse Random Graphs. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 75:1-75:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.75

Author Details

Oleg Verbitsky
  • Institut für Informatik, Humboldt-Universität zu Berlin, Germany
Maksim Zhukovskii
  • School of Computer Science, University of Sheffield, UK

Funding

  • Verbitsky, Oleg: Supported by DFG grant KO 1053/8-2. On leave from the IAPMM, Lviv, Ukraine.

Acknowledgements

The authors would like to thank Michael Krivelevich for helpful discussions.

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