Simplicial Complexes with Many Facets are Vertex Decomposable
Suppose $\Delta$ is a pure simplicial complex on $n$ vertices having dimension $d$ and let $c = n-d-1$ be its codimension in the simplex. Terai and Yoshida proved that if the number of facets of $\Delta$ is at least $\binom{n}{c}-2c+1$, then $\Delta$ is Cohen-Macaulay. We improve this result by showing that these hypotheses imply the stronger condition that $\Delta$ is vertex decomposable. We give examples to show that this bound is optimal, and that the conclusion cannot be strengthened to the class of matroids or shifted complexes. We explore an application to Simon's Conjecture and discuss connections to other results from the literature.
Authors
- Adam Van Tuyl (ORCID: https://orcid.org/0000-0002-6799-6653)
- Ritika Nair (ORCID: https://orcid.org/0000-0003-1413-1696)
- Russ Woodroofe (ORCID: https://orcid.org/0000-0002-8199-3483)
- Jay Schweig (ORCID: https://orcid.org/0000-0001-7529-5454)
- Anton Dochtermann
Institutions
- University of Primorska (SI)
- University of Kansas (US)
- Oklahoma State University Oklahoma City (US)
- McMaster University (CA)
Publication Details
- Journal
- The Electronic Journal of Combinatorics
- Published
- 2024-11-15
- DOI
- https://doi.org/10.37236/12984
- Citations
- 1
- Primary Topic
- Topological and Geometric Data Analysis
- Type
- article
- Field-Weighted Citation Impact
- 0.36
Funders
- Javna Agencija za Raziskovalno Dejavnost RS
- Banff International Research Station for Mathematical Innovation and Discovery
- Natural Sciences and Engineering Research Council of Canada