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.
Highlights
We present several known results on chromatic polynomials of mixed hypergraphs, which will be applied
We define an operation on mixed hypergraphs, which is based on the operation of identifying vertices in graphs
We present our main results after defining some concepts regarding mixed hypergraphs that are necessary to derive the results of this paper
Summary
It is well known that chromatic polynomials of graphs have no real zeros in the intervals (−∞, 0), (0, 1) and (1, 32/27] (see [6,24,25]), while their zeros are dense in the whole complex plane, as explained by Sokal [26]. Whether these properties can be inherited by hypergraphs and mixed hypergraphs has become a natural research question. We further extend the result obtained in [14] on chromatic polynomials of hypergraphs to that on chromatic polynomials of mixed hypergraphs
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