Circular and uniquely colorable mixed hypergraphs

Abstract

A mixed hypergraph is a triple H = (X, C,D), where X is the vertex set, |X| = n, and each of C, D is a family of subsets of X, the C-edges and Dedges, respectively. A proper k-coloring of a mixed hypergraph is a mapping from the vertex set to a set of k colors so that each C-edge has two vertices with a common color and each D-edge has two vertices with distinct colors. A mixed hypergraph is k-colorable (uncolorable; uniquely colorable) if it has a proper coloring with at most k colors (admits no colorings; admits precisely one coloring apart from the permutation of colors). A strict k-coloring is a proper k-coloring when all k colors are used. The minimum number of colors in a strict coloring of H is called the lower chromatic number χ(H); the maximum number is called the upper chromatic number χ(H).