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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