Canonical labelling of random regular graphs

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.