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Canonical labelling of random regular graphs

Mikhail Isaev, Tamás Makai, Brendan McKay, Pawel Pralat, Jane Tan, Maksim Zhukovskii

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.

Canonical labelling of random regular graphs

Abstract

We prove that whenever and as , then with high probability for any non-trivial initial colouring, the colour refinement algorithm distinguishes all vertices of the random regular graph . This, in particular, implies that with high probability admits a canonical labelling computable in time , where is the matrix multiplication exponent.
Paper Structure (21 sections, 22 theorems, 124 equations)

This paper contains 21 sections, 22 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Related work.
  3. Proof strategy.
  4. Organisation.
  5. Notation.
  6. Preliminaries
  7. Concentration Inequalities
  8. Properties of Binomials
  9. Counting Graphs
  10. Sandwiching Graphs
  11. Expansion Properties
  12. Proof of \ref{['thm:main_many_large_set']}: Dense Case
  13. Proof of \ref{['thm:main_many_large_set']}: Sparse Case
  14. Anti-concentration Results
  15. Proof of \ref{['thm:sparse_many_large_set']}
  16. ...and 6 more sections

Key Result

Theorem 1

Let $d_0$ be large enough, let $d=d(n)$ be such that $d_0\leq d\leq n/2$, and let $\mathbf{G}_n \sim \mathcal{G}_{n,d}$. Then, the following holds whp: for every non-trivial partition $[n]=V_1\sqcup V_2$ of the vertex set of $\mathbf{G}_n$, CR runs at most $2\operatorname{diam} (\mathbf{G}_n)+3$ ste

Theorems & Definitions (51)

  • Theorem 1
  • Remark 2
  • Corollary 3
  • Theorem 4
  • Theorem 5
  • Remark 6
  • Lemma 7
  • proof
  • Corollary 8: Anti-concentration of hypergeometric distribution
  • proof
  • ...and 41 more