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

About perfection of circular mixed hypergraphs

A mixed hypergraph is a triple H = (X,C,D), where X is the vertex set and each of C and D is a family of subsets of X, the C-edges and D-edges, respectively. A proper k-coloring of H is a mapping c : X → {1,...,k} such that each C-edge has two vertices with a common color and each D-edge has two vertices with different colors. Maximum number of colors in a coloring using all the colors is called upper chromatic number χ ̄(H). Maximum cardinality of subset of vertices which contains no C-edge is C-stability number αC (H). A mixed hypergraph is called C-perfect if χ ̄ (H') = αC (H') for any induced subhypergraph H'. A mixed hyper- graph H is called circular if there exists a host cycle on the vertex set X such that every edge (C- or D-) induces a connected subgraph on the host cycle. We give a characterization of C-perfect circular mixed hypergraphs

About uniquely colorable mixed hypertrees

A mixed hypergraph is a triple = (X,,) where X is the vertex set and each of , is a family of subsets of X, the -edges and -edges, respectively. A k-coloring of is a mapping c: X → [k] such that each -edge has two vertices with the same color and each -edge has two vertices with distinct colors. = (X,,) is called a mixed hypertree if there exists a tree T = (X,) such that every -edge and every -edge induces a subtree of T. A mixed hypergraph is called uniquely colorable if it has precisely one coloring apart from permutations of colors. We give the characterization of uniquely colorable mixed hypertrees