Chromatic polynomials of some sunflower mixed hypergraphs

The theory of mixed hypergraph coloring was first introduced by Voloshin in 1993 and has been growing ever since. The proper coloring of a mixed hypergraph H = (X, C,D) is the coloring of the vertex set X so that no D-hyperedge is monochromatic and no C-hyperedge is polychromatic. A mixed hypergraph with hyperedges of type D, C or B is commonly known as a D, C, or B-hypergraph respectively where B = C = D. D-hypergraph colorings are the classic hypergraph colorings which have been widely studied. The chromatic polynomial P (H, λ) of a mixed hypergraph H is the function that counts the number of proper λ-colorings, which are mappings f : X → {1, 2, . . . , λ}. A sunflower (hypergraph) with l petals and a core S is a collection of sets e1, . . . , el such that ei ∩ ej = S for all i 6= j. Recently, Walter (see [14]) published some results concerning the chromatic polynomial of some non-uniform D-sunflower. In this paper, we present an alternative proof of his result and extend his formula to those of non-uniform C-sunflowers and B-sunflowers. Some results for a new but related member of sunflowers are also presented. 1. Definitions and notations For basic definitions of graphs and hypergraphs we refer the reader to [1, 4, 12, 15]. A hypergraph H of order n is an ordered pair H=(X, E) where |X| = n is a finite nonempty set of vertices and E is a collection of not necessarily distinct nonempty subsets of X called hyperedges. H is said to be k-uniform if the size of each of its hyperedges is exactly k. A hypergraph is said to be linear if each pair of hyperedges has at most one vertex in common. If the alternating sequence of vertices and distinct hyperedges v0, e1, v1, . . . , el, vl is a hyperpath of length l ≥ 2, then the hypergraph induced by the sequence of hyperedges e1, . . . , el when v0 = vl is called a hypercycle of length l. A hypergraph in which no set of hyperedges induce a hypercycle is said to be acyclic. In this paper all hypergraphs are assumed to be connected and acyclic. Received by the editors August 12, 2012, and in revised form January 22, 2014. 2010 Mathematics Subject Classification. 05C15, 05C30.