Minimal examples of Cohen-Macaulay independence complexes that fail to be shellable or vertex decomposable
Year: 2026
Type: preprint
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 ${\rm Ind}(G)$ is Cohen-Macaulay but not shellable, when the characteristic is not two, and exactly six graphs where ${\rm Ind}(G)$ is shellable but not vertex decomposable.
Source: arXiv (Cornell University)
Author: Van Tuyl, Adam
Institution: McMaster University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220958060