A remark on the Whitney Broken Circuit Theorem

TL;DR

This paper reveals that the Whitney Broken Circuit Theorem is a specific instance of a broader identity linking the chromatic polynomial of a graph to sums over forests, using the connection with the Potts model.

Contribution

It demonstrates that the Whitney Broken Circuit Theorem is a special case of a more general identity involving chromatic polynomials and forests in graphs.

Findings

Whitney Broken Circuit Theorem is a special case of a general identity.

Connection established between chromatic polynomial and Potts model.

General identity relates graph forests to chromatic polynomial.

Abstract

In the present note we show, via the connection between chromatic polynomial and Potts model, that the Whitney Broken circuit theorem is in fact a special case of a more general identity relating the chromatic polynomial of a graph G=(V,E) to sums over forests of G associated to some partition scheme in G.