Monomial bases for broken circuit complexes

Authors:
Jason I. Brown;Bruce E. Sagan
Affiliations:
Department of Mathematics and Statistics, Dalhousie University, Halifax, NS B3H 3J5, Canada;Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA
Venue:
European Journal of Combinatorics
Year:
2009

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Abstract

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.