Chromatic spectrum is broken

Abstract

A mixed hypergraph is a triple H = (X, C,D), where X is the vertex set and each of C, D is a family of subsets of X , the C-edges and D-edges, respectively. A proper k-coloring of a mixed hypergraph is a function 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 if it has a proper coloring with at most k colors. A strict k-coloring is a proper k-coloring using all k colors. The minimum number of colors in a strict coloring of H is its lower chromatic number χ(H); the maximum number is its upper chromatic number χ(H).