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Completing shellings with d−2 extra vertices
Abstract
Simon's conjecture (1994) asserts that any pure d-dimensional shellable complex on n vertices can be extended to the d-skeleton of the simplex on n vertices, one facet at a time, while maintaining shellability. Bolognini and Sentinelli (2026) recently disproved it for every d ≥ 3. We show that the conjecture becomes true once d−2 new vertices are allowed, and that the same holds with k-decomposability in place of shellability for any k ≥ 1. We also show that the number d−2 cannot be lowered. Inflating the counterexample of Bolognini and Sentinelli gives, for every d ≥ 3, a $1$-decomposable complex that cannot be extended in this way with only d−3 new vertices, even while merely maintaining shellability.
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BibTeX
@misc{oh2026completing,
title = {{Completing shellings with $d-2$ extra vertices}},
author = {SuHo Oh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.32721},
eprint = {2609.32721},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32721},
}
RIS
TY - UNPB
TI - Completing shellings with d−2 extra vertices
AU - SuHo Oh
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2609.32721
UR - https://arxiv.org/abs/2609.32721
AB - Simon's conjecture (1994) asserts that any pure d-dimensional shellable complex on n vertices can be extended to the d-skeleton of the simplex on n vertices, one facet at a time, while maintaining shellability. Bolognini and Sentinelli (2026) recently disproved it for every d ≥ 3. We show that the conjecture becomes true once d−2 new vertices are allowed, and that the same holds with k-decomposability in place of shellability for any k ≥ 1. We also show that the number d−2 cannot be lowered. Inflating the counterexample of Bolognini and Sentinelli gives, for every d ≥ 3, a $1$-decomposable complex that cannot be extended in this way with only d−3 new vertices, even while merely maintaining shellability.
ER -