Facet-Hamiltonicity

💡 Research Summary

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The paper introduces a new notion for polytopes called facet‑Hamiltonicity. A facet‑Hamiltonian cycle (or path) in the 1‑skeleton of an n‑dimensional polytope is a closed walk that visits every (n‑1)‑dimensional face (facet) exactly once, with the intersection of the walk and each facet being non‑empty and connected. This can be interpreted as a perfect watch‑man route that guards the entire surface of a simple polytope while using the minimum possible number of edges.

The authors first establish elementary properties of such cycles in simple polytopes. Because each vertex of a simple n‑polytope is incident to n facets, every time a new vertex is entered a new facet is entered and a different facet is left. Consequently, the length of a facet‑Hamiltonian cycle equals the number of facets k, unless the polytope possesses universal facets (facets adjacent to all others), in which case the length can be reduced by the number of universal facets. This observation underlies the later constructions.

The main constructive results are as follows.

1. Permutahedra. The (n‑1)‑dimensional permutahedron has vertices corresponding to all permutations of