PreprintCC BY 4.0arXiv
Minimal examples of Cohen-Macaulay independence complexes that fail to be shellable or vertex decomposable
Abstract
We report on an exhaustive census of all 1,006,700,565 connected graphs on $11$ vertices to find the minimal examples of graphs whose independence complexes are Cohen-Macaulay but not shellable, or shellable but not vertex decomposable. In particular, there are exactly two graphs G where Ind(G) is Cohen-Macaulay but not shellable, when the characteristic is not two, and exactly six graphs where Ind(G) is shellable but not vertex decomposable.
Related
- One-periodic category of twisted complexes, Cohen-Macaulay modules and differential modules
- Field independence of the first seven Betti numbers of flag complexes
- The rational homology of independence complexes of Kneser graphs on triples
- The Homotopy Types of the Independence and Perfect Matching Complexes of Möbius and Circular Ladder Graphs
- An EL-shellable rank-uniform poset that is not UMEL-shellable
BibTeX
@misc{tuyl2026minimal,
title = {{Minimal examples of Cohen-Macaulay independence complexes that fail to be shellable or vertex decomposable}},
author = {Adam Van Tuyl},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09926},
eprint = {2610.09926},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09926},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Minimal examples of Cohen-Macaulay independence complexes that fail to be shellable or vertex decomposable
AU - Adam Van Tuyl
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09926
UR - https://arxiv.org/abs/2610.09926
AB - We report on an exhaustive census of all 1,006,700,565 connected graphs on $11$ vertices to find the minimal examples of graphs whose independence complexes are Cohen-Macaulay but not shellable, or shellable but not vertex decomposable. In particular, there are exactly two graphs G where Ind(G) is Cohen-Macaulay but not shellable, when the characteristic is not two, and exactly six graphs where Ind(G) is shellable but not vertex decomposable.
ER -