The Chromatic Spectrum of Mixed Hypergraphs

A mixed hypergraph is a triple H = (X, C,D), where X is the vertex set, and each of C, D is a list of subsets of X. A strict k-coloring of H is a surjection c : X → {1, . . . , k} such that each member of C has two vertices assigned a common value and each member of D has two vertices assigned distinct values. The feasible set of H is {k : H has a strict k-coloring}. Among other results, we prove that a finite set of positive integers is the feasible set of some mixed hypergraph if and only if it omits the number 1 or is an interval starting with 1. For the set {s, t} with 2 ≤ s ≤ t − 2, the smallest realization has 2t − s vertices. When every member of C ∪ D is a single interval in an underlying linear order on the vertices, the feasible set is also a single interval of integers.