Monomial bases for broken circuit complexes

Let FF be a field and let GG be a finite graph with a total ordering on its edge set. Richard Stanley noted that the Stanley–Reisner ring F(G)F(G) of the broken circuit complex of GG is Cohen–Macaulay. Jason Brown gave an explicit description of a homogeneous system of parameters for F(G)F(G) in terms of fundamental cocircuits in GG. So F(G)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 GG 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 GG from above.