The broken-circuit complex

Abstract

Abstract. The broken-circuit complex introduced by H. Wilf (Which polynomials are chromatic!, Proc. Colloq. Combinational Theory (Rome, 1973)) of a matroid G is shown to be a cone over a related complex, the reduced broken-circuit complex Q'(G). The topological structure of Q'(G) is studied, its Euler characteristic is computed, and joins and skeletons are shown to exist in the class of all such complexes. These computations and constructions are compared with analogous results in the theory of the independent set complex of a matroid. Reduced broken-circuit complexes are partially characterized by conditions concerning which subcomplexes are pure (i.e., have equicardinal maximal simplices). In particular, every matroid complex is a reduced broken-circuit complex. Three proofs are given that the simplex numbers of the cone of Q'(G) are the coefficients which appear in the characteristic polynomial x(G) of G. This relates the work of W. Tutte on externally inactive bases to that of H. Whitney on sets which do not contain broken circuits. One of these proofs gives a combinatorial correspondence between these sets. Properties of the characteristic polynomial are then given topological proofs.

The Tutte polynomial of a morphism of matroids I. Set-pointed matroids and matroid perspectives

“…Hence The coefficients of the chromatic polynomial of a graph have been expressed in terms of broken circuits by H. Whitney. This result has been generalized to matroids by Brylawski [6]. A similar result holds for matroid perspectives.…”

“…The proof of Theorem 8.1 is by deletion/contraction of the greatest element. It generalizes the proof given by Brylawski in the matroid case [6]. (8.4) Suppose ee X.Letx e E\X and C be a circuit of M contained in X U {x}.…”

Coloring complexes and arrangements

“…Let χ M (t) = (t − 1) 3 (t − 2)(t − 8)(t − 10). Then χ M (t) is the characteristic polynomial of the direct sum of 2 coloops and the parallel connection of a 3-point line, 9-point line, and an 11-point line [Br,Cor. 4.7].…”

On cohomology algebras of complex subspace arrangements

“…The subcomplex of I(L) formed by all non-empty sets of atoms that do not contain a broken circuit is called the broken circuit complex BC(L). See [Bj1,Bry] for additional information.…”