Independence complexes of well-covered circulant graphs

TL;DR

This paper investigates the combinatorial and topological properties of independence complexes of well-covered circulant graphs, identifying conditions for various structural properties and providing a comprehensive classification for small graphs.

Contribution

It characterizes when independence complexes of well-covered circulant graphs are shellable, Cohen-Macaulay, or vertex decomposable, and provides a complete classification for graphs with up to 16 vertices.

Findings

Identified conditions for shellability and Cohen-Macaulayness in independence complexes.

Provided a complete table of properties for all well-covered circulant graphs up to 16 vertices.

Discovered an example of a shellable but not vertex decomposable independence complex.

Abstract

We study the independence complexes of families of well-covered circulant graphs discovered by Boros-Gurvich-Milani\v{c}, Brown-Hoshino, and Moussi. Because these graphs are well-covered, their independence complexes are pure simplicial complexes. We determine when these pure complexes have extra combinatorial (e.g. vertex decomposable, shellable) or topological (e.g. Cohen-Macaulay, Buchsbaum) structure. We also provide a table of all well-covered circulant graphs on 16 or less vertices, and for each such graph, determine if it is vertex decomposable, shellable, Cohen-Macaulay, and/or Buchsbaum. A highlight of this search is an example of a graph whose independence complex is shellable but not vertex decomposable.