On the upper chromatic number of a hypergraph

We introduce the notion of a of a hypergraph, which is a subset of vertices to be colored so that at least two vertices are of the same color. Hypergraphs with both and are called mixed hypergraphs. The maximal number of colors for which there exists a mixed hypergraph coloring using all the colors is called the upper chromatic number of a hypergraph H and is denoted by X(H). An algorithm for computing the number of colorings of a mixed hypergraph is proposed. The properties of the upper chromatic number and the colorings of some classes of hypergraphs are discussed. A greedy polynomial time algorithm for finding a lower bound for x( H) of a hypergraph H containing only co-edges is The cardinality of a maximum stable set of an all-vertex partial hypergraph generated by co-edges is called the co-stability number a A (H). A hypergraph H is called co-perfect if x( HI) = a.A (HI) for all its wholly-edge subhypergraphs H' . Two classes of minimal non co-perfect hypergraphs (the so called monostars and cycloids C;r-l, r 3) are found. It is proved that hypertrees are co-perfect if and only if they do not contain monostars as wholly-edge subhypergraphs. It is conjectured that the r-uniform hypergraph H is co-perfect if and only if it contains neither monostars nor cycloids C;r-l) r ~ 3, as whollyedge subhypergraphs.