Cohen-Macaulay graphs and face vectors of flag complexes

TL;DR

This paper introduces a construction linking flag complexes and their face vectors, providing new characterizations of Cohen-Macaulay and Buchsbaum complexes, and explores the relationship between face vectors and $h$-vectors.

Contribution

It presents a novel construction that generates Cohen-Macaulay flag complexes from face vectors, and characterizes bipartite graphs with Cohen-Macaulay independence complexes.

Findings

The construction produces vertex-decomposable, Cohen-Macaulay complexes.

Face vectors of flag complexes correspond to $h$-vectors of certain complexes.

Proves the conjecture for bipartite graphs with Cohen-Macaulay complexes.

Abstract

We introduce a construction on a flag complex that, by means of modifying the associated graph, generates a new flag complex whose $h$-factor is the face vector of the original complex. This construction yields a vertex-decomposable, hence Cohen-Macaulay, complex. From this we get a (non-numerical) characterisation of the face vectors of flag complexes and deduce also that the face vector of a flag complex is the $h$-vector of some vertex-decomposable flag complex. We conjecture that the converse of the latter is true and prove this, by means of an explicit construction, for $h$-vectors of Cohen-Macaulay flag complexes arising from bipartite graphs. We also give several new characterisations of bipartite graphs with Cohen-Macaulay or Buchsbaum independence complexes.