The face lattice of any simplicial polytope is Hamiltonian
Abstract
We show that the cover graph of the face lattice of every simplicial polytope admits a Hamilton cycle. Moreover, any two distinct facets can be prescribed as the neighbors of the full polytope on this cycle. By duality, the face lattice of every simple polytope is Hamiltonian as well. This proves the simplicial and simple cases of a conjecture of Behrooznia, Brenner, Merino, Mütze, Rieck, and Verciani (SODA26). The proof uses a line shelling in both directions. The forward shelling partitions the faces into Boolean intervals, whose cover graphs are cubes. The reverse shelling gives a second partition, on which we construct a perfect matching by adding or removing a fixed vertex in each interval. This matching connects the cubes of the first partition. A theorem of Gregor then completes it to a Hamilton cycle using only edges within these cubes. The same argument applies to shellable simplicial spheres, although without the prescribed neighbors of the full complex in this case. Using Chari's shelling construction of perfect matchings, we also obtain Hamiltonicity for polytopes with at most one facet that is not a simplex.
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BibTeX
@misc{lauff2026face,
title = {{The face lattice of any simplicial polytope is Hamiltonian}},
author = {Robert Lauff},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19938},
eprint = {2609.19938},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19938},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - The face lattice of any simplicial polytope is Hamiltonian
AU - Robert Lauff
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19938
UR - https://arxiv.org/abs/2609.19938
AB - We show that the cover graph of the face lattice of every simplicial polytope admits a Hamilton cycle. Moreover, any two distinct facets can be prescribed as the neighbors of the full polytope on this cycle. By duality, the face lattice of every simple polytope is Hamiltonian as well. This proves the simplicial and simple cases of a conjecture of Behrooznia, Brenner, Merino, Mütze, Rieck, and Verciani (SODA26). The proof uses a line shelling in both directions. The forward shelling partitions the faces into Boolean intervals, whose cover graphs are cubes. The reverse shelling gives a second partition, on which we construct a perfect matching by adding or removing a fixed vertex in each interval. This matching connects the cubes of the first partition. A theorem of Gregor then completes it to a Hamilton cycle using only edges within these cubes. The same argument applies to shellable simplicial spheres, although without the prescribed neighbors of the full complex in this case. Using Chari's shelling construction of perfect matchings, we also obtain Hamiltonicity for polytopes with at most one facet that is not a simplex.
ER -