Generalized Whitney formulas for broken circuits in ambigraphs and matroids

TL;DR

This paper generalizes Whitney's theorem for chromatic polynomials and symmetric functions to broader settings including ambigraphs, hypergraphs, and matroids, using sign-reversing involutions for proofs.

Contribution

It introduces new generalizations of Whitney's formulas applicable to ambigraphs and matroids, extending classical graph theory results to more complex combinatorial structures.

Findings

Whitney's formula holds under various generalizations.

A simple sign-reversing involution proves these generalized formulas.

Matroids can replace graphs in the chromatic polynomial context.

Abstract

We explore several generalizations of Whitney's theorem -- a classical formula for the chromatic polynomial of a graph. Following Stanley, we replace the chromatic polynomial by the chromatic symmetric function. Following Dohmen and Trinks, we exclude not all but only an (arbitrarily selected) set of broken circuits, or even weigh these broken circuits with weight monomials instead of excluding them. Following Crew and Spirkl, we put weights on the vertices of the graph. Following Gebhard and Sagan, we lift the chromatic symmetric function to noncommuting variables. In addition, we replace the graph by an "ambigraph", an apparently new concept that includes both hypergraphs and multigraphs as particular cases. We show that Whitney's formula endures all these generalizations, and a fairly simple sign-reversing involution can be used to prove it in each setting. Furthermore, if we restrict ourselves to the chromatic polynomial, then the graph can be replaced by a matroid. We discuss an application to transitive digraphs (i.e., posets), and reprove an alternating-sum identity by Dahlberg and van Willigenburg.