Cohen–Macaulay Graphs and Face Vectors of Flag 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-vector is the face vector of the original complex. This construction yields a vertex-decomposable, hence Cohen–Macaulay, complex. From this we get a (nonnumerical) characterization 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 characterizations of bipartite graphs with Cohen–Macaulay or Buchsbaum independence complexes.