A characterization of uniquely colorable mixed hypergraphs of order n with upper chromatic numbers n-1 and n-2

A mixed hypergraph consists of two families of subsets of the vertex set: the V-edges and the C-edges. In a suitable coloring of a mixed hypergraph, every C-edge has at least two vertices of the same color, and every V-edge has at least two vertices colored differently. The largest and smallest possible numbers of colors in a coloring are called the upper and lower chromatic numbers, X and X, respectively. A mixed hypergraph is uniquely colorable if it has just one coloring apart from permutations of colors. We characterize all uniquely colorable mixed hypergraphs of order n with X(1-l) = X(1-l) = n 1 and n 2.