Reconstructing simplicial spheres from facet adjacency
The facet-ridge graph $\operatorname{FR}(K)$ of a finite simplicial complex $K$ homeomorphic to $S^d$ has the $d$-dimensional faces of $K$ as vertices, two of them adjacent when they share a $(d-1)$-dimensional face. Blind and Mani proved that for boundary complexes of simplicial polytopes every isomorphism of facet-ridge graphs is induced by a unique simplicial isomorphism (in dual form: a simple polytope is determined by its graph), and Kalai gave a constructive proof. Every simplicial $2$-sphere is the boundary complex of a $3$-polytope, by Steinitz's theorem. Ceballos and Doolittle proved the extension property for the spherical subword complexes of finite Coxeter groups, and Yang proved that every shellable simplicial sphere is determined by its facet-ridge graph. Ceballos and Doolittle also gave two non-isomorphic triangulations of the torus, and two of the real projective plane, with the same facet-ridge graph.
Problem. Let $d \ge 3$ and let $K$ and $L$ be finite simplicial complexes homeomorphic to $S^d$. Is every graph isomorphism $\varphi : \operatorname{FR}(K) \to \operatorname{FR}(L)$ induced by a unique simplicial isomorphism $K \to L$? No shellability or polytopality is assumed.
Source: Blind and Mani (1987), Question 1, conjectured by Kalai (2009); in this form Ceballos and Doolittle (2025), Conjecture 1.2. Stated for d ≥ 3 by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- CO — Combinatorics
- Secondary
- AT, MG
- Keywords
- AIM Open Applied Problems, facet-ridge graphs, polytope reconstruction, simplicial spheres
References
@article{blind1987,
author = {Blind, Roswitha and Mani-Levitska, Peter},
title = {Puzzles and polytope isomorphisms},
journal = {Aequationes Math.},
volume = {34},
number = {2-3},
year = {1987},
pages = {287--297},
doi = {10.1007/BF01830678},
}
@misc{kalai2009,
author = {Kalai, Gil},
title = {Telling a simple polytope from its graph},
howpublished = {Blog post, Combinatorics and more},
year = {2009},
url = {https://gilkalai.wordpress.com/2009/01/16/telling-a-simple-polytope-from-its-graph/},
}
@article{ceballos2025,
author = {Ceballos, Cesar and Doolittle, Joseph},
title = {Subword complexes and {K}alai's conjecture on reconstruction of spheres},
journal = {Discrete Comput. Geom.},
volume = {74},
number = {1},
year = {2025},
pages = {23--48},
doi = {10.1007/s00454-025-00733-6},
eprint = {2206.15461},
archivePrefix = {arXiv},
}
@article{kalai1988,
author = {Kalai, Gil},
title = {A simple way to tell a simple polytope from its graph},
journal = {J. Combin. Theory Ser. A},
volume = {49},
number = {2},
year = {1988},
pages = {381--383},
doi = {10.1016/0097-3165(88)90064-7},
}
@misc{yang2024,
author = {Yang, Yirong},
title = {Reconstructing a shellable sphere from its facet-ridge graph},
year = {2024},
eprint = {2401.04220},
archivePrefix = {arXiv},
}
@misc{aim-kalai-sphere-reconstruction,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Reconstructing simplicial spheres from facet adjacency}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/618-kalai-sphere-reconstruction.md},
note = {Entry 618 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00872, "Reconstructing simplicial spheres from facet adjacency", clopen.solutions.