# Recursion relations for chromatic coefficients for graphs and hypergraphs

## TL;DR

This paper develops recursion relations for chromatic polynomial coefficients of graphs and hypergraphs, generalizes Whitney's broken cycle theorem, and provides explicit formulas for specific hypergraph coefficients.

## Contribution

It introduces new recursion relations and extends classical theorems to hypergraphs, along with explicit formulas for hypergraph chromatic coefficients.

## Key findings

- Established recursion relations for chromatic coefficients
- Generalized Whitney's broken cycle theorem to hypergraphs
- Derived explicit formulas for hypergraph chromatic coefficients

## Abstract

We establish a set of recursion relations for the coefficients in the chromatic polynomial of a graph or a hypergraph. As an application we provide a generalization of Whitney's broken cycle theorem for hypergraphs, as well as deriving an explicit formula for the linear coefficient of the chromatic polynomial of the $r$-complete hypergraph in terms of roots of the Taylor polynomials for the exponential function.