Chromatic Polynomials of Some Mixed Hypergraphs

Motivated by a recent result of M. Walter [Electron. J. Comb. 16, No. 1, Research Paper R94, 16 p. (2009; Zbl 1186.05059)] concerning the chromatic polynomials of some hypergraphs, we present the chromatic polynomials of several (non-uniform) mixed hypergraphs. We use a recursive process for generating explicit formulae for linear mixed hypercacti and multi-bridge mixed hypergraphs using a decomposition of the underlying hypergraph into blocks, defined via chains. Further, using an algebra software package such as Maple, one can use the basic formulae and process demonstrated in this paper to generate the chromatic polynomials for any linear mixed hypercycle, unicyclic mixed hypercyle, mixed hypercactus and multi-bridge mixed hypergraph. We also give the chromatic polynomials of several examples in illustration of the process including the formulae for some mixed sunflowers

Chromatic polynomials of some sunflower mixed hypergraphs

The theory of mixed hypergraphs coloring has been first introduced by Voloshin in 1993 and it 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. Recently, Walter published [15] 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 of a new but related member of sunflowers are also presented

Chromatic polynomials of some sunflower mixed hypergraphs

The theory of mixed hypergraphs coloring has been first introduced by Voloshin in 1993 and it has been growing ever since. The proper coloring of a mixed hypergraph H = (X; C;D) is the coloring ofthe 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-hypergraphrespectively where B = C = D. D-hypergraph colorings are the classichypergraph colorings which have been widely studied. The chro-matic polynomial P(H;) of a mixed hypergraph H is the function thatcounts the number of proper ??colorings, which are mappings f : X !f1; 2; : : : ; g. A sunfower (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 published [14] some results concerning the chromatic polynomial of some non-uniform D-sunfower. In this paper, we present an alternative proof of his result and extend his formula to those of non-uniform C-sunowers and B-sunowers. Some results of a new but related member of sunfowers are also presented

Chromatic Numbers of Simplicial Manifolds

Higher chromatic numbers $\chi_s$ of simplicial complexes naturally generalize the chromatic number $\chi_1$ of a graph. In any fixed dimension $d$, the $s$-chromatic number $\chi_s$ of $d$-complexes can become arbitrarily large for $s\leq\lceil d/2\rceil$ [6,18]. In contrast, $\chi_{d+1}=1$, and only little is known on $\chi_s$ for $\lceil d/2\rceil<s\leq d$. A particular class of $d$-complexes are triangulations of $d$-manifolds. As a consequence of the Map Color Theorem for surfaces [29], the 2-chromatic number of any fixed surface is finite. However, by combining results from the literature, we will see that $\chi_2$ for surfaces becomes arbitrarily large with growing genus. The proof for this is via Steiner triple systems and is non-constructive. In particular, up to now, no explicit triangulations of surfaces with high $\chi_2$ were known. We show that orientable surfaces of genus at least 20 and non-orientable surfaces of genus at least 26 have a 2-chromatic number of at least 4. Via a projective Steiner triple systems, we construct an explicit triangulation of a non-orientable surface of genus 2542 and with face vector $f=(127,8001,5334)$ that has 2-chromatic number 5 or 6. We also give orientable examples with 2-chromatic numbers 5 and 6. For 3-dimensional manifolds, an iterated moment curve construction [18] along with embedding results [6] can be used to produce triangulations with arbitrarily large 2-chromatic number, but of tremendous size. Via a topological version of the geometric construction of [18], we obtain a rather small triangulation of the 3-dimensional sphere $S^3$ with face vector $f=(167,1579,2824,1412)$ and 2-chromatic number 5.Comment: 22 pages, 11 figures, revised presentatio

Chromatic Polynomials of Mixed Hypercyles

We color the vertices of each of the edges of a C-hypergraph (or cohypergraph) in such a way that at least two vertices receive the same color and in every proper coloring of a B-hypergraph (or bihypergraph), we forbid the cases when the vertices of any of its edges are colored with the same color (monochromatic) or when they are all colored with distinct colors (rainbow). In this paper, we determined explicit formulae for the chromatic polynomials of C-hypercycles and B-hypercycles

Chromatic Polynomials Of Some (m,l)-Hyperwheels

In this paper, using a standard method of computing the chromatic polynomial of hypergraphs, we introduce a new reduction theorem which allows us to find explicit formulae for the chromatic polynomials of some (complete) non-uniform $(m,l)-$hyperwheels and non-uniform $(m,l)-$hyperfans. These hypergraphs, constructed through a ``join" graph operation, are some generalizations of the well-known wheel and fan graphs, respectively. Further, we revisit some results concerning these graphs and present their chromatic polynomials in a standard form that involves the Stirling numbers of the second kind

On Cohen-Macaulay Hopf monoids in species

We study Cohen-Macaulay Hopf monoids in the category of species. The goal is to apply techniques from topological combinatorics to the study of polynomial invariants arising from combinatorial Hopf algebras. Given a polynomial invariant arising from a linearized Hopf monoid, we show that under certain conditions it is the Hilbert polynomial of a relative simplicial complex. If the Hopf monoid is Cohen- Macaulay, we give necessary and sufficient conditions for the corresponding relative simplicial complex to be relatively Cohen-Macaulay, which implies that the polynomial has a nonnegative h-vector. We apply our results to the weak and strong chromatic polynomials of acyclic mixed graphs, and the order polynomial of a double poset

On Cohen-Macaulay Hopf monoids in species

We study Cohen-Macaulay Hopf monoids in the category of species. The goal is to apply techniques from topological combinatorics to the study of polynomial invariants arising from combinatorial Hopf algebras. Given a polynomial invariant arising from a linearized Hopf monoid, we show that under certain conditions it is the Hilbert polynomial of a relative simplicial complex. If the Hopf monoid is Cohen-Macaulay, we give necessary and sufficient conditions for the corresponding relative simplicial complex to be relatively Cohen-Macaulay, which implies that the polynomial has a nonnegative $h$-vector. We apply our results to the weak and strong chromatic polynomials of acyclic mixed graphs, and the order polynomial of a double poset.Comment: 13 pages, extended abstract for FPSAC 202