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

Formulas for the computation of the Tutte polynomial of graphs with parallel classes

We give some reduction formulas for computing the Tutte polynomial of any graph with parallelclasses. Several examples are given to illustrate our results

On the Choosability of Some Graphs

Suppose ch(G) and X(G) denote, respectively, the choice number and the chromatic number of a graph G = (V;E). If ch(G) = (G) then G is said to be chromatic-choosable. Recently, Reed et al. proved a conjecture by Ohba that states that G is chromatic choosable whenever jV (G)j <= 2(G) + 1. Here, we present otherclasses of chromatic-choosable graphs that do not satisfy the hypothesis of the proven conjecture. Moreover, we give the upper bounds for the choice numbers of the Mycielski graphs and the cartesian product of any two graphs, in terms of a vertex-neighborhood condition

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

Tutte Polynomial of Multi-Bridge Graphs

In this paper, using a well-known recursion for computing the Tutte polynomial of any graph, we found explicit formulae for the Tutte polynomials of any multi-bridge graph and some $2-$tree graphs. Further, several recursive formulae for other graphs such as the fan and the wheel graphs are also discussed

On the Existence of S-graphs

We answer in the affirmative a question posed by S. Al-Addasi and H. Al-Ezeh in [Int. J. Math. Math. Sci. 2008, Article ID 468583, 11 p. (2008; Zbl 1161.05316)] on the existence of symmetric diametrical bipartite graphs of diameter 4. Bipartite symmetric diametrical graphs are calls S-graphs by some authors and diametrical graphs have also been studied by other authors using different terminology, such as self-centered unique eccentric point graphs. We include a brief survey of some of this literature and advertise that the existence question was answered by A. Berman and A. Kotzig [Ann. Discrete Math. 8, 37–42 (1980; Zbl 0446.05025)], along with a study of different isomorphism classes of these graphs using a (1,-1)-matrix representation which includes the well-known Hadamard matrices. Our presentation focuses on a neighborhood characterization of S-graphs and we conclude our survey with a beautiful version of this characterization known to T. N. Janakiraman [Discrete Math. 126, No. 1–3, 411–414 (1994; Zbl 0792.05117)]]]> 2012 English http://libres.uncg.edu/ir/ecsu/f/Allagan-S-graphs_AHJS.pdf oai:libres.uncg.edu/38718 2023-02-19T20:48:38Z ECSU Hall numbers of some complete k-partite graphs Allagan , Julian A. D. NC DOCKS at Elizabeth City State University <![CDATA[The Hall number is a graph parameter closely related to the choice number. Here it is shown that the Hall numbers of the complete multipartite graphs K(m,2,?,2), m=2, are equal to their choice numbers

F-WORM colorings of some 2-trees: partition vectors

A collection of distinct subgraphs of a graphG = (V;E).An F-WORM coloring of G is the coloring of its vertices such that no copy of each subgraphFi 2 F is monochrome or rainbow. This generalizes the notion of F-WORMcoloring that was introduced recently by W. Goddard, K. Wash, and H. Xu. A (restricted)partition vector is a sequence whose terms r are the number of F-WORMcolorings using exactly r colors, with. The partition vectors of complete graphsand those of some 2-trees are discussed. We show that, although 2-trees admit the samepartition vector in classic proper vertex colorings which forbid monochrome K2, their partition vectors in K3-WORM colorings are different

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