An Abstraction of Whitney's Broken Circuit Theorem

TL;DR

This paper generalizes Whitney's broken circuit theorem to a wide class of combinatorial invariants, unifying and extending known results across graph theory, matroid theory, and lattice theory.

Contribution

It introduces a broad abstraction of Whitney's theorem applicable to various polynomials and invariants, including new generalizations and unified proofs.

Findings

Unified framework for chromatic, domination, and subgraph polynomials

New generalizations of classical combinatorial identities

Expanded applications to matroid invariants and convex geometries

Abstract

We establish a broad generalization of Whitney's broken circuit theorem on the chromatic polynomial of a graph to sums of type $\sum_{A\subseteq S} f(A)$ 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 M\"obius function of a finite lattice, which generalizes Rota's crosscut theorem. For the classical M\"obius 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).