Zero-free intervals of chromatic polynomials of hypergraphs

Zero-Free Intervals of Chromatic Polynomials of Mixed Hypergraphs

“…In this article, we further extend the result obtained in [14] on chromatic polynomials of hypergraphs to that on chromatic polynomials of mixed hypergraphs.…”

“…Let L 0 be the set of hypergraphs satisfying conditions (I) and (II) and L 0 be the set of hypergraphs satisfying conditions (I), (II), and (IV). In [14], the first two authors of this article obtained the following results on hypergraphs in the two sets L 0 and L 0 .…”

“…Although this approach has not been realized within the past century, the theory of chromatic polynomials has evolved into an important and independent branch of graph theory, encouraging substantial numbers of studies in this field [2][3][4][5][6][7][8][9][10][11]. In the 1960s, the concept of chromatic polynomials was extended to hypergraphs [12][13][14][15][16][17][18][19][20][21]. In 1993, Voloshin [22] introduced the concept of mixed hypergraphs as well as their chromatic polynomials, which are actually generalizations of chromatic polynomials of graphs and hypergraphs.…”

“…The study of the zero distributions of chromatic polynomials is an active research topic [6][7][8][9][10][11]13,14,23]. For graphs, 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].…”

“…Back in 1995, Dohmen [12] showed that for any hypergraph in which each edge has an even size and each cycle contains an edge of size 2, it chromatic polynomial is zero-free in the interval (−∞, 0). In 2017, the first two authors of this article in [13] proved that chromatic polynomials of hypergraphs have dense real zeros in set of all real numbers, while they [14] recently extended Dohmen's result to a larger family of hypergraphs and proved that the existence of families of hypergraphs whose chromatic polynomials are zero-free in the intervals (−∞, 0) and (0, 1).…”