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 CohenMacaulay. 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.