Canonical labelling of sparse random graphs

TL;DR

This paper presents a polynomial-time algorithm for canonical labeling of sparse Erdős-Rényi random graphs when p=O(1/n), improving understanding of automorphism groups and graph isomorphism in this regime.

Contribution

The authors develop a new efficient algorithm combining color refinement and tree canonization for sparse random graphs, extending prior results to a broader class.

Findings

High-probability canonical labeling for p=O(1/n)

Algorithm runs in O(n log n) time

Complete description of automorphism groups of the 2-core

Abstract

We show that if $p=O(1/n)$, then the Erd\H{o}s-R\'{e}nyi random graph $G(n,p)$ with high probability admits a canonical labeling computable in time $O(n\log n)$. Combined with the previous results on the canonization of random graphs, this implies that $G(n,p)$ with high probability admits a polynomial-time canonical labeling whatever the edge probability function $p$. Our algorithm combines the standard color refinement routine with simple post-processing based on the classical linear-time tree canonization. Noteworthy, our analysis of how well color refinement performs in this setting allows us to complete the description of the automorphism group of the 2-core of $G(n,p)$.