Year: 2020
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Kiefer, Sandra, McKay, Brendan D.
Institutions: RWTH Aachen University, Australian National University
Language: English
Topic: Graph Theory and Algorithms
Subfield: Computer Vision and Pattern Recognition
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3026872109