A Classification of Long-Refinement Graphs for Colour Refinement

TL;DR

This paper characterizes graphs that require the maximum number of Colour Refinement iterations, providing a complete classification for low degrees and insights into their structure, with implications for graph isomorphism testing.

Contribution

It offers a full classification of long-refinement graphs with degrees up to 4, using a reverse-engineering approach and structural analysis, filling a key gap in understanding Colour Refinement complexity.

Findings

Long-refinement graphs with degrees 2 and 3 are fully characterized.

All low-degree long-refinement graphs can be represented as compact strings.

Graphs distinguished only in the last iteration of Colour Refinement do not exist.

Abstract

The Colour Refinement algorithm is a classical procedure to detect symmetries in graphs, whose most prominent application is in graph-isomorphism tests. The algorithm and its generalisation, the Weisfeiler-Leman algorithm, evaluate local information to compute a colouring for the vertices in an iterative fashion. Different final colours of two vertices certify that no isomorphism can map one onto the other. The number of iterations that the algorithm takes to terminate is its central complexity parameter. For a long time, it was open whether graphs that take the maximum theoretically possible number of Colour Refinement iterations actually exist. Starting from an exhaustive search on graphs of low degrees, Kiefer and McKay proved the existence of infinite families of such long-refinement graphs with degrees 2 and 3, thereby showing that the trivial upper bound on the iteration number of Colour Refinement is tight. In this work, we provide a complete characterisation of the long-refinement graphs with low (or, equivalently, high) degrees. We show that, with one exception, the aforementioned families are the only long-refinement graphs with maximum degree at most 3, and we fully classify the long-refinement graphs with maximum degree 4. To this end, via a reverse-engineering approach, we show that all low-degree long-refinement graphs can be represented as compact strings, and we derive multiple structural insights from this surprising fact. Since long-refinement graphs are closed under taking edge complements, this also yields a classification of long-refinement graphs with high degrees. Kiefer and McKay initiated a search for long-refinement graphs that are only distinguished in the last iteration of Colour Refinement before termination. We conclude it in this submission by showing that such graphs cannot exist.