Monomial bases for broken circuit complexes

Let F be a field and let G be a finite graph with a total ordering on its edge set. Richard Stanley noted that the Stanley–Reisner ring F ( G ) of the broken circuit complex of G is Cohen–Macaulay. Jason Brown gave an explicit description of a homogeneous system of parameters for F ( G ) in terms of fundamental cocircuits in G . So F ( G ) modulo this hsop is a finite dimensional vector space. We conjecture an explicit monomial basis for this vector space in terms of the circuits of G and prove that the conjecture is true for two infinite families of graphs. We also explore an application of these ideas to bounding the number of acyclic orientations of G from above.