Canonical labelling of random regular graphs

TL;DR

This paper proves that for large random regular graphs with increasing degree, the color refinement algorithm can distinguish all vertices, enabling a canonical labelling computable efficiently, which advances graph isomorphism understanding.

Contribution

It establishes that the color refinement algorithm distinguishes all vertices in large random regular graphs with high probability, leading to efficient canonical labelling.

Findings

Color refinement distinguishes all vertices in large random regular graphs.

Canonical labelling can be computed in near-optimal time for these graphs.

Results hold when degree and number of vertices grow unbounded.

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 $O(\min\{n^{\omega},nd^2+nd\log n\})$, where $\omega<2.372$ is the matrix multiplication exponent.