Zero-Free Intervals of Chromatic Polynomials of Mixed Hypergraphs

Abstract

A mixed hypergraph H is a triple (X,C,D), where X is a finite set and each of C and D is a family of subsets of X. For any positive integer λ, a proper λ-coloring of H is an assignment of λ colors to vertices in H such that each member in C contains at least two vertices assigned the same color and each member in D contains at least two vertices assigned different colors. The chromatic polynomial of H is the graph-function counting the number of distinct proper λ-colorings of H whenever λ is a positive integer. In this article, we show that chromatic polynomials of mixed hypergraphs under certain conditions are zero-free in the intervals (−∞,0) and (0,1), which extends known results on zero-free intervals of chromatic polynomials of graphs and hypergraphs.

Keywords

• mixed hypergraph
• chromatic polynomial
• zero-free interval