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arXiv · 1509.08251

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

Christoph Berkholz·Paul Bonsma·Martin Grohe
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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.

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Christoph Berkholz, Paul Bonsma, Martin Grohe. 2015-09-28. Tight Lower and Upper Bounds for the Complexity of Canonical Colour Refinement. https://arxiv.org/abs/1509.08251

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