Tight Lower and Upper Bounds for the Complexity of Canonical Colour Refinement

TL;DR

This paper presents an optimal algorithm with tight bounds for canonical colour refinement in graphs, crucial for graph isomorphism testing, establishing both upper and lower bounds on complexity.

Contribution

It introduces an $O((m+n)\log n)$ algorithm for canonical colour refinement and proves this bound is tight under certain assumptions, advancing understanding of graph isomorphism complexity.

Findings

The algorithm runs in $O((m+n)\log n)$ time.

No faster algorithm exists under the considered assumptions.

The bounds are tight, matching the best known algorithms.

Abstract

An assignment of colours to the vertices of a graph is stable if any two vertices of the same colour have identically coloured neighbourhoods. The goal of colour refinement is to find a stable colouring that uses a minimum number of colours. This is a widely used subroutine for graph isomorphism testing algorithms, since any automorphism needs to be colour preserving. We give an $O((m+n)\log n)$ algorithm for finding a canonical version of such a stable colouring, on graphs with $n$ vertices and $m$ edges. We show that no faster algorithm is possible, under some modest assumptions about the type of algorithm, which captures all known colour refinement algorithms.