The Iteration Number of Colour Refinement

TL;DR

This paper investigates the iteration complexity of the Colour Refinement algorithm, proving that the trivial upper bound of n-1 iterations is tight by constructing graphs that require nearly this many steps to stabilize.

Contribution

It provides explicit constructions of graphs demonstrating that Colour Refinement can take as many as n-1 iterations, establishing the tightness of the known upper bound.

Findings

Existence of graphs requiring n-1 iterations for stabilization

Construction of graphs requiring at least n-2 iterations for n >= 10

Confirmation that the upper bound on iteration number is tight

Abstract

The Colour Refinement procedure and its generalisation to higher dimensions, the Weisfeiler-Leman algorithm, are central subroutines in approaches to the graph isomorphism problem. In an iterative fashion, Colour Refinement computes a colouring of the vertices of its input graph. A trivial upper bound on the iteration number of Colour Refinement on graphs of order n is n-1. We show that this bound is tight. More precisely, we prove via explicit constructions that there are infinitely many graphs G on which Colour Refinement takes |G|-1 iterations to stabilise. Modifying the infinite families that we present, we show that for every natural number n >= 10, there are graphs on n vertices on which Colour Refinement requires at least n-2 iterations to reach stabilisation.