Cohen-Macaulay graphs and face vectors of flag complexes

Details Cohen-Macaulay graphs and face vectors of flag complexes Cohen-Macaulay graphs and face vectors of flag complexes

2010-03-23

arxiv.org/abs/1003.4447v3

SIAM J. Discrete Math. 26 (2012), no. 1, 89--101

Mathematics

Combinatorics

14 pages, 3 figures; major update

Scientific paper

10.1137/100818170

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.

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