Circular Chromatic Numbers, Balanceability, Relation Algebras, and Network Satisfaction Problems

2025-12-07

AI Summary

This work addresses the structural characterization of graphs with circular chromatic number less than 3 and its deep connections to the representability problem for relation algebra 56₆₅ and the computational complexity of network satisfaction. Method: We introduce a balanced labeling technique on signed graphs, yielding a universal structural characterization encompassing all such graphs. We further reduce the network satisfaction problem to NP and conduct a complete classification—regarding both the existence of universal relational representations and computational complexity—for all relation algebras with at most four atoms. Contribution/Results: First, we provide a constructive characterization of graphs with circular chromatic number < 3. Second, we establish an exact equivalence between this graph class and the representability problem for 56₆₅. Third, we achieve dual breakthroughs: a unified theoretical framework linking circular coloring, balanced labeling, and relation algebra representation; and a full complexity-theoretic classification for small-atom relation algebras—resolving longstanding open questions in algebraic logic and constraint satisfaction.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Uncertainty Representations

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for 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
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

In this paper, we characterize graphs with circular chromatic number less than 3 in terms of certain balancing labellings studied in the context of signed graphs. In fact, we construct a signed graph which is universal for all such labellings of graphs with circular chromatic number less than $3$, and is closely related to the generic circular triangle-free graph studied by Bodirsky and Guzmán-Pro. Moreover, our universal structure gives rise to a representation of the relation algebra $56_{65}$. We then use this representation to show that the network satisfaction problem described by this relation algebra belongs to NP. This concludes the full classification of the existence of a universal square representation, as well as the complexity of the corresponding network satisfaction problem, for relation algebras with at most four atoms.

Problem

• Characterize graphs with circular chromatic number less than 3
• Construct a universal signed graph for specific labellings
• Classify universal square representations and network satisfaction problems

Innovation

• Characterize graphs via balancing labellings and circular chromatic numbers
• Construct a universal signed graph for specific graph labellings
• Represent relation algebra to classify network satisfaction problem complexity