Zero-free intervals of chromatic polynomials of mixed hypergraphs

Ruixue Zhang, Fengming Dong

Published 2018-12-05Version 1

A mixed hypergraph $\mathcal{H}=(\mathcal{V}, \mathcal{C}, \mathcal{D})$ consists of a vertex set $\mathcal{V}$ and two subsets $\mathcal{C}$ and $\mathcal{D}$ of $\{E\subseteq \mathcal{V}: |E|\ge 1\}$. For any positive integer $\lambda$, a proper $\lambda$-coloring of $\mathcal{H}$ is an assignment of $\lambda$ colors to vertices in $\mathcal{H}$ such that each member in $\mathcal{C}$ contains at least two vertices assigned the same color and each member in $\mathcal{D}$ contains at least two vertices assigned different colors. The chromatic polynomial of $\mathcal{H}$, denoted by $P(\mathcal{H}, \lambda)$, is the function which counts the number of proper $\lambda$-colorings of $\mathcal{H}$ whenever $\lambda$ is a positive integer. In this paper, we show that $P(\mathcal{H}, \lambda)$ is zero-free in the intervals $(-\infty, 0)$ and $(0, 1)$ under certain conditions. This result extends known results on zero-free intervals of the chromatic polynomials of graphs and hypergraphs.