On the upper chromatic number of a hypergraph ∗ Vitaly I. Voloshin Department of Mathematics & Cybernetics, Moldova State University, A. Mateevich str., 60, Kishinev, 277009, Moldova email: vol@usm.md Abstract. We introduce the notion of a co-edge 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 edges and co-edges are called mixed hypergraphs. The maximal number of colors for which there ex- ists a mixed hypergraph coloring using all the colors is called the upper chromatic number of a hypergraph H and is denoted by ¯ χ(H ). An al- gorithm for computing the number of colorings of a mixed hypergraph is proposed. The properties of the upper chromatic number and the color- ings of some classes of hypergraphs are discussed. A greedy polynomial time algorithm for finding a lower bound for ¯ χ(H ) of a hypergraph H containing only co-edges is presented. The cardinality of a maximum stable set of an all-vertex partial hy- pergraph generated by co-edges is called the co-stability number α A (H ). A hypergraph H is called co-perfect if ¯ χ(H ′ )= α A (H ′ ) for all its wholly- edge subhypergraphs H ′ . Two classes of minimal non co-perfect hyper- graphs (the so called monostars and cycloids C r 2r−1 ,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 2r−1 ,r ≥ 3, as wholly-edge subhypergraphs. 1. Introduction. Let X = {x 1 ,x 2 ,...,x n } be a set of sources of power supply such that the action time of any source is one quantum of time and all sources acting for any given quantum of time switch on and switch off synchronously. Consider the following general constraints on their common work: 1) let A = {A 1 ,A 2 ,...,A k },A i ⊆ X, i =1,...,k, k ≥ 1, be a family of subsets of X such that at least two sources from every A i act for the same quantum of time; * Australasian Journal of Combinatorics, No 11, 1995, p.25-45; while revising the paper the author was partially supported by Volkswagen–Stiftung Project No.I/69041 1