Properties of chromatic polynomials of hypergraphs not held for chromatic polynomials of graphs

TL;DR

This paper explores unique properties of hypergraph chromatic polynomials, revealing that they have all integers as zeros, dense real zeros, and roots with multiplicities that differ from those of graph chromatic polynomials.

Contribution

It demonstrates novel properties of hypergraph chromatic polynomials, including their zeros and root multiplicities, which do not occur in graph chromatic polynomials.

Findings

Chromatic polynomials of hypergraphs have all integers as zeros.

They contain dense real zeros in the set of real numbers.

Root multiplicities can be greater than one for hypergraph chromatic polynomials.

Abstract

In this paper, we present some properties on chromatic polynomials of hypergraphs which do not hold for chromatic polynomials of graphs. We first show that chromatic polynomials of hypergraphs have all integers as their zeros and contain dense real zeros in the set of real numbers. We then prove that for any multigraph $G=(V,E)$, the number of totally cyclic orientations of $G$ is equal to the value of $|P(H,-1)|$, where $P(H,\lambda)$ is the chromatic polynomial of a hypergraph $H$ which is constructed from $G$. Finally we show that the multiplicity of root "$0$" of $P(H,\lambda)$ may be at least $2$ for some connected hypergraphs $H$, and the multiplicity of root "$1$" of $P(H,\lambda)$ may be $1$ for some connected and separable hypergraphs $H$ and may be $2$ for some connected and non-separable hypergraphs $H$.