Uniquely colorable mixed hypergraphs

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. ∗ Supported by COBASE † Supported in part by the Hungarian Scientific Research Fund through grants OTKA T–026575 and T–032969 ‡ Partially supported by Volkswagen–Stiftung Project No.I/69041, COBASE short and long-term grants (Georgia State University and University of Illinois at Urbana-Champaign), and CNR (University of Catania); while revising the paper partially supported by DFG (TU-Dresden)