PreprintCC BY 4.0arXiv
An EL-shellable rank-uniform poset that is not UMEL-shellable
Abstract
We address Questions~8.1 and~8.4 of Coron--Ferroni--Li \cite{coron2025chow}. For the first part of Question~8.1, which asks whether every rank-uniform EL-shellable poset admits a rank-uniform EL-labeling, we construct a counterexample of rank $4$ and prove that this rank is minimal. For the second part of Question~8.1, concerning whether every rank-uniform EL-shellable poset is UMEL-shellable, we construct a counterexample of rank $3$. We also develop a general criterion for constructing UMEL-shellable posets of rank $3$. As an application, we give an affirmative answer to Question~8.4 of Coron--Ferroni--Li.
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BibTeX
@misc{aithan2026elshellable,
title = {{An EL-shellable rank-uniform poset that is not UMEL-shellable}},
author = {Hadi Al Aithan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30323},
eprint = {2609.30323},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30323},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An EL-shellable rank-uniform poset that is not UMEL-shellable
AU - Hadi Al Aithan
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.30323
UR - https://arxiv.org/abs/2609.30323
AB - We address Questions~8.1 and~8.4 of Coron--Ferroni--Li \cite{coron2025chow}. For the first part of Question~8.1, which asks whether every rank-uniform EL-shellable poset admits a rank-uniform EL-labeling, we construct a counterexample of rank $4$ and prove that this rank is minimal. For the second part of Question~8.1, concerning whether every rank-uniform EL-shellable poset is UMEL-shellable, we construct a counterexample of rank $3$. We also develop a general criterion for constructing UMEL-shellable posets of rank $3$. As an application, we give an affirmative answer to Question~8.4 of Coron--Ferroni--Li.
ER -