Uniquely colorable mixed hypergraphs Original Research Article

A mixed hypergraph consists of two families of edges: the C-edges and D-edges. In a coloring, every C-edge has at least two vertices of the same color, while every D-edge has at least two vertices colored differently. The largest and smallest possible numbers of colors in a coloring are termed the upper and lower chromatic number, χ̄ and χ, respectively. A mixed hypergraph is called uniquely colorable if it has precisely one coloring apart from the permutation of colors. We begin a systematic study of uniquely colorable mixed hypergraphs. In particular, we show that every colorable mixed hypergraph can be embedded into some uniquely colorable mixed hypergraph; we investigate the role of uniquely colorable subhypergraphs being separators, study recursive operations (orderings and subset contractions) and unique colorings, and prove that it is NP-hard to decide whether a mixed hypergraph is uniquely colorable. We also discuss the weaker property where the mixed hypergraph has a unique coloring with χ̄ colors and a unique coloring with χ colors, where χ̄>χ. The class of these “weakly uniquely colorable” mixed hypergraphs contains all uniquely colorable graphs in the usual sense.