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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


The broken-circuit complex

Author: Tom Brylawski
Journal: Trans. Amer. Math. Soc. 234 (1977), 417-433
MSC: Primary 05B35; Secondary 05B25, 05C15
MathSciNet review: 468931
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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 $ \mathcal{C}'(G)$. The topological structure of $ \mathcal{C}'(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 $ \mathcal{C}'(G)$ are the coefficients which appear in the characteristic polynomial $ \chi (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.

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PII: S 0002-9947(1977)0468931-6
Keywords: Basis, broken circuit, broken-circuit complex, characteristic polynomial, chromatic polynomial, combinatorial geometry, Euler characteristic, external activity, graph, internal activity, matroid, pure complex, simplex numbers, simplicial complex, Whitney polynomial
Article copyright: © Copyright 1977 American Mathematical Society