Monomial Bases for NBC Complexes

In a seminal paper, Richard Stanley noted that the Stanley- Reisner ring R of an NBC (no broken circuit) complex of a graph G is Cohen- Macaulay. Jason Brown gave an explicit description of a homogeneous system of parameters for R in terms of fundamental edge-cuts in G. So R modulo this h.s.o.p. is a finite dimensional vector space. We conjecture an explicit monomial basis for this vector space in terms of the circuts of G and prove that the conjecture is true for several families of graphs.