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Documents authored by McKay, Brendan D.

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2026.114
Canonical Labelling of Random Regular Graphs

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

Published in: LIPIcs, Volume 374, 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)

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

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)

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@InProceedings{isaev_et_al:LIPIcs.ICALP.2026.114,
  author =	{Isaev, Mikhail and Makai, Tam\'{a}s and McKay, Brendan D. and Pra{\l}at, Pawe{\l} and Tan, Jane and Zhukovskii, Maksim},
  title =	{{Canonical Labelling of Random Regular Graphs}},
  booktitle =	{53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026)},
  pages =	{114:1--114:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-428-4},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{374},
  editor =	{Bhattacharya, Sayan and Nanongkai, Danupon and Benedikt, Michael and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.114},
  URN =		{urn:nbn:de:0030-drops-265039},
  doi =		{10.4230/LIPIcs.ICALP.2026.114},
  annote =	{Keywords: random graphs, regular graphs, colour refinement, canonical labelling, graph isomorphism}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2020.73
The Iteration Number of Colour Refinement

Authors: Sandra Kiefer and Brendan D. McKay

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Abstract
The Colour Refinement procedure and its generalisation to higher dimensions, the Weisfeiler-Leman algorithm, are central subroutines in approaches to the graph isomorphism problem. In an iterative fashion, Colour Refinement computes a colouring of the vertices of its input graph. A trivial upper bound on the iteration number of Colour Refinement on graphs of order n is n-1. We show that this bound is tight. More precisely, we prove via explicit constructions that there are infinitely many graphs G on which Colour Refinement takes |G|-1 iterations to stabilise. Modifying the infinite families that we present, we show that for every natural number n ≥ 10, there are graphs on n vertices on which Colour Refinement requires at least n-2 iterations to reach stabilisation.
Cite as

Sandra Kiefer and Brendan D. McKay. The Iteration Number of Colour Refinement. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 73:1-73:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kiefer_et_al:LIPIcs.ICALP.2020.73,
  author =	{Kiefer, Sandra and McKay, Brendan D.},
  title =	{{The Iteration Number of Colour Refinement}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{73:1--73:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.73},
  URN =		{urn:nbn:de:0030-drops-124801},
  doi =		{10.4230/LIPIcs.ICALP.2020.73},
  annote =	{Keywords: Colour Refinement, iteration number, Weisfeiler-Leman algorithm, quantifier depth}
}
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