An Abstraction of Whitney's Broken Circuit Theorem. Dohmen, K. and Trinks, M. April, 2014.
abstract   bibtex   
We establish a broad generalization of Whitney's broken circuit theorem on the chromatic polynomial of a graph to sums of type $∑_A⊂eq S f(A)$ where $S$ is a finite set and $f$ is a mapping from the power set of $S$ into an abelian group. We give applications to the domination polynomial and the subgraph component polynomial of a graph, the chromatic polynomial of a hypergraph, the characteristic polynomial and Crapo's beta invariant of a matroid, and the principle of inclusion-exclusion. Thus, we discover several known and new results in a concise and unified way. As further applications of our main result, we derive a new generalization of the maximums-minimums identity and of a theorem due to Blass and Sagan on the Möbius function of a finite lattice, which generalizes Rota's crosscut theorem. For the classical Möbius function, both Euler's totient function and its Dirichlet inverse, and the reciprocal of the Riemann zeta function we obtain new expansions involving the greatest common divisor resp. least common multiple. We finally establish an even broader generalization of Whitney's broken circuit theorem in the context of convex geometries (antimatroids).
@misc{ DT:2014:AWBCT,
  author = {Klaus Dohmen and Martin Trinks},
  title = {An Abstraction of {Whitney}'s Broken Circuit Theorem},
  year = {2014},
  month = {April},
  pubstate = {Submitted for publication},
  eprinttype = {arXiv},
  eprint = {1404.5480},
  eprintclass = {math-CO},
  abstract = {We establish a broad generalization of Whitney's broken
                  circuit theorem on the chromatic polynomial of a graph to
                  sums of type $∑_{A⊂eq S} f(A)$ where $S$ is a
                  finite set and $f$ is a mapping from the power set of $S$
                  into an abelian group. We give applications to the
                  domination polynomial and the subgraph component polynomial
                  of a graph, the chromatic polynomial of a hypergraph, the
                  characteristic polynomial and Crapo's beta invariant of a
                  matroid, and the principle of inclusion-exclusion.  Thus, we
                  discover several known and new results in a concise and
                  unified way.  As further applications of our main result, we
                  derive a new generalization of the maximums-minimums
                  identity and of a theorem due to Blass and Sagan on the
                  Möbius function of a finite lattice, which generalizes
                  Rota's crosscut theorem.  For the classical Möbius
                  function, both Euler's totient function and its Dirichlet
                  inverse, and the reciprocal of the Riemann zeta function we
                  obtain new expansions involving the greatest common divisor
                  resp. least common multiple.  We finally establish an even
                  broader generalization of Whitney's broken circuit theorem
                  in the context of convex geometries (antimatroids).},
  keywords = {graph, hypergraph, matroid, chromatic polynomial,
                  domination polynomial, subgraph component polynomial,
                  characteristic polynomial, beta invariant, broken circuit,
                  broken neighbourhood, inclusion-exclusion, Möbius
                  function, lattice, maximum-minimums identity, totient,
                  Dirichlet inverse, Riemann zeta function, closure system,
                  convex geometry}
}
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