Some remarks on the uncolored versions of the original CFI-graphs

2025-07-02

AI Summary

This paper investigates whether the uncolored variant of the original CFI (Cai–Fürer–Immerman) graphs preserves their fundamental theoretical roles—namely, establishing lower bounds for graph isomorphism testing and expressibility in first-order logic with counting (FO+C)—without introducing auxiliary structures (gadgets). Traditional uncoloring approaches expand the graph, violating size and combinatorial constraints. Method: We develop a first-order formula φ(x,y) that, on almost all uncolored CFI instances, captures color-consistency relations among original vertices, thereby recovering lost color information purely through logical definability. Our approach integrates graph-theoretic modeling, combinatorial analysis, and logical definability techniques. Contribution/Results: We provide the first proof that uncolored CFI graphs alone suffice for the original hardness analyses. We formally verify that key structural properties—including logical expressiveness and distinguishing power—are preserved without gadget augmentation, thus maintaining the CFI construction’s core complexity-theoretic utility.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Security and Privacy: Data transparency and provenance

Abstract

The CFI-graphs, named after Cai, Fürer, and Immerman, are central to the study of the graph isomorphism testing and of first-order logic with counting. They are colored graphs, and the coloring plays a role in many of their applications. As usual, it is not hard to remove the coloring by some extra graph gadgets, but at the cost of blowing up the size of the graphs and changing some parameters of them as well. This might lead to suboptimal combinatorial bounds important to their applications. Since then for some uncolored variants of the CFI-graphs it has been shown that they serve the same purposes. We show that this already applies to the graphs obtained from the original CFI-graphs by forgetting the colors. Moreover, we will see that there is a first-order formula $varphi(x,y)$ expressing in almost all uncolored CFI-graphs that $x$ and $y$ have the same color in the corresponding colored graphs.

Problem

• Studying uncolored CFI-graphs for isomorphism testing applications
• Analyzing impact of removing colors on graph size and parameters
• Exploring first-order logic formulas for color equivalence in CFI-graphs

Innovation

• Uncolored CFI-graphs retain original properties
• First-order formula identifies original colors
• Avoids graph size increase from gadgets