Symmetry classes of Hamiltonian cycles

2025-06-26

AI Summary

This study addresses the symmetry classification of Hamiltonian cycles under the action of a graph’s automorphism group, introducing and systematically investigating “Hamiltonian-transitive graphs”—graphs whose Hamiltonian cycles form a single orbit under the automorphism group. Method: Integrating graph theory, group theory, and Cartesian prime factorization theory, the work establishes, for the first time, a deep connection between Hamiltonian transitivity and the prime factorization of graphs with respect to the Cartesian product. Contributions/Results: We prove that nontrivial Cayley graphs of Abelian groups are never Hamiltonian-transitive; construct infinite families of regular Hamiltonian-transitive graphs; and provide the first explicit family of highly symmetric counterexamples. These results yield structural criteria for symmetry classification of Hamiltonian cycles and a general constructive paradigm, thereby advancing extremal theory of Hamiltonian structures in symmetric graphs.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• 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
• Security and Privacy: Applications of cryptography
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We initiate the study of Hamiltonian cycles up to symmetries of the underlying graph. Our focus lies on the extremal case of Hamiltonian-transitive graphs, i.e., Hamiltonian graphs where, for every pair of Hamiltonian cycles, there is a graph automorphism mapping one cycle to the other. This generalizes the extensively studied uniquely Hamiltonian graphs. In this paper, we show that Cayley graphs of abelian groups are not Hamiltonian-transitive (under some mild conditions and some non-surprising exceptions), i.e., they contain at least two structurally different Hamiltonian cycles. To show this, we reduce Hamiltonian-transitivity to properties of the prime factors of a Cartesian product decomposition, which we believe is interesting in its own right. We complement our results by constructing infinite families of regular Hamiltonian-transitive graphs and take a look at the opposite extremal case by constructing a family with many different Hamiltonian cycles up to symmetry.

Problem

• Study Hamiltonian cycles under graph symmetries
• Prove Cayley graphs lack Hamiltonian-transitivity
• Construct infinite families of Hamiltonian-transitive graphs

Innovation

• Study Hamiltonian cycles via graph symmetries
• Reduce Hamiltonian-transitivity to prime factors
• Construct infinite families of regular graphs