Monomial Bases for Broken Circuit Complexes

Details Monomial Bases for Broken Circuit Complexes Monomial Bases for Broken Circuit Complexes

2006-01-25

arxiv.org/abs/math/0601619v1

Mathematics

Combinatorics

Scientific paper

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.

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