A Generalization of NBC Bases to Broken Circuit Complexes of Matroids

TL;DR

This paper extends the concept of NBC bases from graphs to regular matroids, providing a broader framework and identifying classes of matroids with these bases, supported by deletion-contraction principles.

Contribution

It generalizes the NBC basis concept to regular matroids and establishes a deletion-contraction axiom for their existence, expanding the combinatorial understanding.

Findings

Identified two infinite classes of matroids with NBC bases.

Extended the linear system of parameters to regular matroids.

Proved the existence of NBC bases in the generalized setting.

Abstract

Brown has shown that the Stanley-Reisner ring of the broken circuit complex of a graph has a linear system of parameters which is defined in terms of the circuits and cocircuits of the graph. Later on Brown and Sagan conjectured a special set of monomials - a so-called NBC basis - described in terms of the circuits and cocircuits of the graph to be a monomial basis for the corresponding quotient of the Stanley-Reisner ring and proved this to be true for theta and phi graphs. We generalize the aforementioned linear system of parameters to broken circuit complexes of regular matroids and transfer the notion of NBC bases to the general setting of regular matroids. We are able to obtain the analogous results to the ones of Brown and Sagan in this more general context. We show a deletion-contraction axiom for the existence of NBC bases. Using this results we identify two infinite classes of matroids which have NBC bases and which are the matroid theoretic analogue of theta and phi graphs.