A Classification of Long-Refinement Graphs for Colour Refinement

TL;DR

The paper resolves the long-standing question of whether long-refinement graphs exist beyond initial constructions by Kiefer and McKay, delivering a complete classification for graphs with maximum degree at most $4$. It introduces a reverse-engineering framework that recovers all possible refinement histories from late-stage partitions and encodes low-degree structures as compact strings, yielding exhaustive families for degrees $\\{2,3\\}$ and new classifications for degree $4$. By leveraging the fact that long-refinement graphs are closed under edge complementation, the authors extend these insights to graphs with high degrees. They also confirm that no pair of long-refinement graphs can be distinguished only in the final iteration, addressing a natural extension of prior open questions. Overall, the work advances the theoretical understanding of Colour Refinement’s limits and provides a rigorous, structure-driven catalogue of all low-degree long-refinement graphs, with implications for graph isomorphism and related logic/complexity questions.

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.

Figures (16)

• Figure 1: The partitions $\pi^{p-1}$ and $\pi^{p}$, where the splitting order of pairs is from left to right. $P_a$ is blue, $P_b$ is first blue and then pink, respectively. For clarity, we omit all edges between non-consecutive pairs apart from the ones connecting $P_a$ and $P_b$ to the minimum of the linear order.
• Figure 2: The unique long-refinement graph with degrees $\{1,3\}$.
• Figure 3: The long-refinement graphs $G$ with $\deg(G) \neq \{2,3\}$, omitting the infinite families from Tables \ref{['tab:adj-list-d=l-34']} -- \ref{['tab:adj-list3-d=l-34']}.
• Figure 4: $\pi^{p-1}$ when $b < n_\mathcal{P}$. $P_a$ is blue, $P_b$ is pink and blue, respectively. For clarity, we omit all edges between non-consecutive pairs.
• Figure 5: Induction step for the proof of Lemma \ref{['lem:first-symmetry']} in the case that $b = a + 1$. The arrangement of the pairs follows the splitting order $\prec$, i.e. a pair further left has a lower index in $\prec$. The unique new pairs in $\pi^{p-h+1}$ (in pink and brown) are in symmetric position to $P_a$ and $P_b$ (in blue) and united to a class of size $4$ (in brown) in $\pi^{p-h}$.

Theorems & Definitions (77)

• Definition 2.1: Colour Refinement
• Theorem 2.3
• Theorem 2.4
• Corollary 2.5
• Theorem 2.6
• Theorem 2.7
• Theorem 2.8
• Theorem 2.9
• Lemma 2.10
• proof