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Canonization of a random circulant graph by counting walks

Part of: Graph theory

Published online by Cambridge University Press:  26 November 2025

Oleg Verbitsky  and
Maksim Zhukovskii Open the ORCID record for Maksim Zhukovskii [Opens in a new window]
Oleg Verbitsky
Affiliation:
Institut für Informatik, Humboldt-Universität zu Berlin, Berlin, Germany
Maksim Zhukovskii*
Affiliation:
School of Computer Science, University of Sheffield, Sheffield, UK
*
Corresponding author: Maksim Zhukovskii; Email: m.zhukovskii@sheffield.ac.uk
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Abstract

It is well known that almost all graphs are canonizable by a simple combinatorial routine known as colour refinement, also referred to as the 1-dimensional Weisfeiler–Leman algorithm. With high probability, this method assigns a unique label to each vertex of a random input graph and, hence, it is applicable only to asymmetric graphs. The strength of combinatorial refinement techniques becomes a subtle issue if the input graphs are highly symmetric. We prove that the combination of colour refinement and vertex individualization yields a canonical labelling for almost all circulant digraphs (i.e., Cayley digraphs of a cyclic group). This result provides first evidence of good average-case performance of combinatorial refinement within the class of vertex-transitive graphs. Remarkably, we do not even need the full power of the colour refinement algorithm. We show that the canonical label of a vertex $v$ can be obtained just by counting walks of each length from $v$ to an individualized vertex. Our analysis also implies that almost all circulant graphs are compact in the sense of Tinhofer, that is, their polytops of fractional automorphisms are integral. Finally, we show that a canonical Cayley representation can be constructed for almost all circulant graphs by the more powerful 2-dimensional Weisfeiler–Leman algorithm.

Keywords

Canonical labelinggraph isomorphismcirculant graphrandom graphCayley graphwalk countsWeisfeiler-Leman algorithm

MSC classification

Primary: 05C60: Isomorphism problems (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.)
Secondary: 05C80: Random graphs

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 35 , Issue 2 , March 2026 , pp. 178 - 206
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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Footnotes

*

A preliminary version of this paper was presented at WALCOM’24, the 18th International Conference and Workshops on Algorithms and Computation [45].

#

On leave from the IAPMM, Lviv, Ukraine

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