A Counterexample to Prescribed-Cycle Recovery in Barnette Graphs

Bekos, Kaufmann, and Pfister asked whether every prescribed Hamiltonian cycle of a Barnette graph can be recovered by choosing the dual spanning tree in the book-embedding algorithm of Alam et al. We answer this question negatively. An explicit 16-vertex Barnette graph has a Hamiltonian cycle whose complementary perfect matching meets all three edge-color classes in the simultaneous edge/face coloring used by the algorithm. Every output of the algorithm contains every edge designated green, independently of the spanning tree. The prescribed cycle omits a green edge under every global color labeling and therefore cannot be an output.

This record contains the venue-neutral preprint and LaTeX source, a dependency-free certificate verifier, the exact graph/embedding/coloring certificate, and a pinned Lean 4 companion. The Lean companion connects facial incidence to the induced edge colors, checks the displayed Hamiltonian order and exact cycle/matching partition, and certifies connectivity after all deletions of at most two vertices.

The Lean source contains no sorry declarations or custom axioms. Finite propositions using native_decide are explicitly disclosed and accompanied by complete #print axioms audits. The topological interpretation of the oriented incidence data and the published algorithmic Property 1 remain identified external inputs.

AI disclosure: OpenAI Codex assisted with literature and novelty searches, certificate verification, code testing, proof development, and manuscript drafting and editing. Anthropic Claude was used for independent adversarial review. The author independently checked the work, accepts full responsibility, and confirms that neither system is an author.