Zero-Free Intervals of Chromatic Polynomials of Mixed Hypergraphs

Ruixue Zhang, Fengming Dong and Meiqiao Zhang
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Ruixue Zhang: School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China
Fengming Dong: National Institute of Education, Nanyang Technological University, Singapore 637616, Singapore
Meiqiao Zhang: National Institute of Education, Nanyang Technological University, Singapore 637616, Singapore

Mathematics, 2022, vol. 10, issue 2, 1-11

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 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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