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To determine if every prescribed Hamiltonian cycle in Barnette graphs can be recovered using dual spanning trees.

August 13, 2026Open Access

A Counterexample to Prescribed-Cycle Recovery in Barnette Graphs

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To determine if every prescribed Hamiltonian cycle in Barnette graphs can be recovered using dual spanning trees.
Constructed a specific 16-vertex Barnette graph.
Analyzed the edge-coloring methods and outputs of the book-embedding algorithm.
Certified findings using Lean 4 for mathematical verification.
Identified a Hamiltonian cycle that omits a green edge in all global color labelings.
Demonstrated that the algorithm's output will always include the designated green edges.
Concluded that the prescribed cycle cannot be produced by the algorithm.

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Counterexample demonstrates that not all Hamiltonian cycles in Barnette graphs can be recovered as expected.

Lennart Rudolph (2026) studied this question.

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