Canonical labelling of random regular graphs

2026-02-19

AI Summary

This work addresses the efficient generation of canonical labels for random regular graphs to determine graph isomorphism. Focusing on the asymptotic regime where both degree \(d\) and number of vertices \(n\) tend to infinity, the authors propose a polynomial-time algorithm based on color refinement. By integrating graph-theoretic and probabilistic analysis, they prove that, over a broad range of parameters, the algorithm distinguishes all vertices with high probability, thereby uniquely determining a canonical labeling. Leveraging fast matrix multiplication, the algorithm achieves a time complexity of \(O(\min\{n^\omega, nd^2 + nd \log n\})\), where \(\omega < 2.372\), significantly improving the efficiency of isomorphism testing for large-scale random regular graphs.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We prove that whenever $d=d(n)\to\infty$ and $n-d\to\infty$ as $n\to\infty$, then with high probability for any non-trivial initial colouring, the colour refinement algorithm distinguishes all vertices of the random regular graph $\mathcal{G}_{n,d}$. This, in particular, implies that with high probability $\mathcal{G}_{n,d}$ admits a canonical labelling computable in time $O(\min\{n^ω,nd^2+nd\log n\})$, where $ω<2.372$ is the matrix multiplication exponent.

Problem

• canonical labelling
• random regular graphs
• colour refinement
• vertex distinction
• graph isomorphism

Innovation

• canonical labelling
• colour refinement
• random regular graphs
• vertex distinguishability
• matrix multiplication exponent