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Canonical Labelling of Random Regular Graphs

Authors Mikhail Isaev , Tamás Makai , Brendan D. McKay , Paweł Prałat , Jane Tan , Maksim Zhukovskii

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Random graphs
  • Mathematics of computing → Graph algorithms
Keywords
  • random graphs
  • regular graphs
  • colour refinement
  • canonical labelling
  • graph isomorphism

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Abstract

We prove that whenever d = d(n) → ∞ and n-d → ∞ as n → ∞, then with high probability for any non-trivial initial colouring, the colour refinement algorithm distinguishes all vertices of the random regular graph 𝒢_{n,d}. This, in particular, implies that with high probability 𝒢_{n,d} admits a canonical labelling computable in time O(min{n^ω, nd²+ndlog n}), where ω < 2.372 is the matrix multiplication exponent.

Cite As Get BibTex

Mikhail Isaev, Tamás Makai, Brendan D. McKay, Paweł Prałat, Jane Tan, and Maksim Zhukovskii. Canonical Labelling of Random Regular Graphs. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 114:1-114:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.114

Author Details

Mikhail Isaev
  • School of Mathematics and Statistics, UNSW Sydney, Australia
Tamás Makai
  • Institute of Mathematics, LMU Munich, Germany
Brendan D. McKay
  • School of Computing, Australian National University, Canberra, Australia
Paweł Prałat
  • Department of Mathematics, Toronto Metropolitan University, Canada
Jane Tan
  • Mathematical Institute, University of Oxford, UK
Maksim Zhukovskii
  • School of Computer Science, The University of Sheffield, UK

Funding

  • McKay, Brendan D.: Supported by Australian Research Council grant DP190100977.

References

  1. J. Alman, R. Duan, V. Vassilevska Williams, Y. Xu, Z. Xu, and R. Zhou. More asymmetry yields faster matrix multiplication. In Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'25), pages 2005-2039, 2025. Google Scholar
  2. N. Alon. Eigenvalues and expanders. Combinatorica, 6(2):83-96, 1986. URL: https://doi.org/10.1007/BF02579166.
  3. N. Alon and F. R. K. Chung. Explicit construction of linear sized tolerant networks. Discrete Mathematics, 72:15-19, 1988. URL: https://doi.org/10.1016/0012-365X(88)90189-6.
  4. N. Alon, M. Krivelevich, and V. H. Vu. On the concentration of eigenvalues of random symmetric matrices. Israel Journal of Mathematics, 131:259-267, 2002. Google Scholar
  5. M. Anastos, M. Kwan, and B. Moore. Smoothed analysis for graph isomorphism. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing (STOC'25), pages 2098-2106, 2025. Google Scholar
  6. L. Babai. Graph isomorphism in quasipolynomial time. In Proceedings of the 48th Annual ACM Symposium on Theory of Computing (STOC'16), pages 684-697, 2016. Google Scholar
  7. L. Babai. Canonical form for graphs in quasipolynomial time: preliminary report. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC'19), pages 1237-1246, 2019. Google Scholar
  8. L. Babai, P. Erdős, and S. M. Selkow. Random graph isomorphism. SIAM Journal on Computing, 9(3):628-635, 1980. URL: https://doi.org/10.1137/0209047.
  9. L. Babai and E. M. Luks. Canonical labeling of graphs. In Proceedings of the 15th Annual ACM Symposium on Theory of Computing (STOC'83), pages 171-183, 1983. Google Scholar
  10. R. Bauerschmidt, J. Huang, A. Knowles, and H.-T. Yau. Edge rigidity and universality of random regular graphs of intermediate degree. Geometric and Functional Analysis, 30:693-769, 2020. Google Scholar
  11. N. Behague, D. Il'ković, and R. Montgomery. A proof of the Kim-Vu sandwich conjecture. arXiv preprint, 2025. URL: https://arxiv.org/abs/2510.20765.
  12. C. Berkholz, P. Bonsma, and M. Grohe. Tight lower and upper bounds for the complexity of canonical colour refinement. Theory of Computing Systems, 60:581-614, 2017. URL: https://doi.org/10.1007/S00224-016-9686-0.
  13. B. Bollobás. Distinguishing vertices of random graphs. Annals of Discrete Mathematics, 13:33-50, 1982. Google Scholar
  14. B. Bollobás. Random Graphs. Cambridge University Press, 2 edition, 2001. Google Scholar
  15. A. Z. Broder, A. M. Frieze, S. Suen, and E. Upfal. Optimal construction of edge-disjoint paths in random graphs. SIAM Journal on Computing, 28(2):541-573, 1999. URL: https://doi.org/10.1137/S0097539795290805.
  16. N. Chiba and T. Nishizeki. Arboricity and subgraph listing algorithms. SIAM Journal on Computing, 14(1):210-223, 1985. URL: https://doi.org/10.1137/0214017.
  17. N. A. Cook, L. Goldstein, and T. Johnson. Size biased couplings and the spectral gap for random regular graphs. Annals of Probability, 46(1):72-125, 2018. Google Scholar
  18. T. Czajka and G. Pandurangan. Improved random graph isomorphism. Journal of Discrete Algorithms, 6:85-92, 2008. URL: https://doi.org/10.1016/J.JDA.2007.01.002.
  19. A. Dudek, A. Frieze, A. Ruciński, and M. Šileikis. Embedding the Erdős-Rényi hypergraph into the random regular hypergraph and hamiltonicity. Journal of Combinatorial Theory, Series B, 122:719-740, 2017. URL: https://doi.org/10.1016/J.JCTB.2016.09.003.
  20. J. Friedman. A proof of Alon’s second eigenvalue conjecture and related problems. Memoirs of the American Mathematical Society, 195(910), 2008. Google Scholar
  21. P. Gao. Triangles and subgraph probabilities in random regular graphs. Electronic Journal of Combinatorics, 31(1):P1.2, 2024. URL: https://doi.org/10.37236/10281.
  22. P. Gao, M. Isaev, and B. D. McKay. Sandwiching random regular graphs between binomial random graphs. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'20), pages 690-701, 2020. Google Scholar
  23. P. Gao, M. Isaev, and B. D. McKay. Sandwiching dense random regular graphs between binomial random graphs. Probability Theory and Related Fields, 184(1-2):115-158, 2022. Google Scholar
  24. P. Gao, M. Isaev, and B. D. McKay. Kim-Vu’s sandwich conjecture is true for d ≫ log⁴ n. arXiv preprint, 2023. URL: https://arxiv.org/abs/2011.09449.
  25. J. Gaudio, M. Z. Rácz, and A. Sridhar. Average-case and smoothed analysis of graph isomorphism. Annals of Applied Probability, 35(2):1373-1406, 2025. Google Scholar
  26. Y. He. Spectral gap and edge universality of dense random regular graphs. Communications in Mathematical Physics, 405(181):1-40, 2024. Google Scholar
  27. H. A. Helfgott, J. Bajpai, and D. Dona. Graph isomorphisms in quasi-polynomial time. arXiv preprint, 2017. URL: https://arxiv.org/abs/1710.04574.
  28. N. Immerman and E. S. Lander. Describing graphs: A first-order approach to graph canonization. In Complexity Theory Retrospective, pages 59-81. Springer, New York, 1990. Google Scholar
  29. M. Isaev, T. Makai, B. D. McKay, P. Prałat, J. Tan, and M. Zhukovskii. Canonical labelling of random regular graphs. arXiv preprint, 2026. URL: https://arxiv.org/abs/2602.17567.
  30. S. Janson, T. Łuczak, and A. Ruciński. Random Graphs. Wiley, 2000. Google Scholar
  31. J. H. Kim, B. Sudakov, and V. Vu. On the asymmetry of random regular graphs and random graphs. Random Structures & Algorithms, 21(3-4):216-224, 2002. URL: https://doi.org/10.1002/RSA.10054.
  32. J. H. Kim, B. Sudakov, and V. Vu. Small subgraphs of random regular graphs. Discrete Mathematics, 307:1961-1967, 2007. URL: https://doi.org/10.1016/J.DISC.2006.09.032.
  33. J. H. Kim and V. H. Vu. Sandwiching random graphs: universality between random graph models. Advances in Mathematics, 188(2):444-469, 2004. Google Scholar
  34. T. Klimošová, C. Reiher, A. Ruciński, and M. Šileikis. Sandwiching biregular random graphs. Combinatorics, Probability and Computing, 32(1):1-44, 2023. URL: https://doi.org/10.1017/S0963548322000049.
  35. M. Krivelevich and B. Sudakov. Pseudo-random graphs. In More Sets, Graphs and Numbers, volume 15 of Bolyai Society Mathematical Studies, pages 199-262. Springer, 2006. Google Scholar
  36. M. Krivelevich, B. Sudakov, V. Vu, and N. Wormald. Random regular graphs of high degree. Random Structures & Algorithms, 18:346-363, 2001. URL: https://doi.org/10.1002/RSA.1013.
  37. L. Kučera. Canonical labeling of regular graphs in linear average time. In 28th Annual Symposium on Foundations of Computer Science (SFCS'87), pages 271-279, 1987. Google Scholar
  38. A. Liebenau and N. Wormald. Asymptotic enumeration of graphs by degree sequence, and the degree sequence of a random graph. Journal of the European Mathematical Society, 26:1-40, 2024. Google Scholar
  39. N. Linial and J. Mosheiff. On the rigidity of sparse random graphs. Journal of Graph Theory, 85(2):466-480, 2017. URL: https://doi.org/10.1002/JGT.22073.
  40. E. Luks. Isomorphism of graphs of bounded valence can be tested in polynomial time. Journal of Computer and System Sciences, 25:42-65, 1982. URL: https://doi.org/10.1016/0022-0000(82)90009-5.
  41. B. D. McKay. Subgraphs of random graphs with specified degrees. Congressus Numerantium, 33:213-223, 1981. Google Scholar
  42. B. D. McKay and N. C. Wormald. Automorphisms of random graphs with specified degrees. Combinatorica, 4:325-338, 1984. Google Scholar
  43. B. D. McKay and N. C. Wormald. Asymptotic enumeration by degree sequence of graphs of high degree. European Journal of Combinatorics, 11:565-580, 1990. URL: https://doi.org/10.1016/S0195-6698(13)80042-X.
  44. B. D. McKay and N. C. Wormald. Asymptotic enumeration by degree sequence of graphs with degrees o(√ n). Combinatorica, 11:369-382, 1991. Google Scholar
  45. A. Sarid. The spectral gap of random regular graphs. Random Structures & Algorithms, 63(2):557-587, 2023. URL: https://doi.org/10.1002/RSA.21150.
  46. K. Tikhomirov and P. Youssef. The spectral gap of dense random regular graphs. Annals of Probability, 41(1):362-419, 2019. Google Scholar
  47. O. Verbitsky and M. Zhukovskii. Canonical labeling of sparse random graphs. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS'25), pages 75:1-75:20, 2025. Google Scholar
  48. V. Vu. Random discrete matrices. In Horizon of Combinatorics, volume 17 of Bolyai Society Mathematical Studies, pages 257-280. Springer, 2008. URL: https://doi.org/10.1007/978-3-540-77200-2.
  49. B. Weisfeiler and A. Leman. The reduction of a graph to canonical form and the algebra which appears therein. Nauchno-Technicheskaya Informatsia, 9(2):12-16, 1968. Google Scholar
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