The Chromatic Spectrum of Mixed Hypergraphs

Abstract.

 A mixed hypergraph is a triple ℋ=(X, ?, ?), where X is the vertex set, and each of ?, ? 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 ? has two vertices assigned a common value and each member of ? 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 ?∪? is a single interval in an underlying linear order on the vertices, the feasible set is also a single interval of integers.