The mixed hypergraphs

We introduce the notion of an anti-edge of a hypergraph, which is a non-overall polychromatic subset of vertices. The maximal number of colors, for which there exists a coloring of a hy-pergraph using all colors, is called an upper chromatic number of a hypergraph H and denoted by ¯ χ(H). The general algorithm for computing the numbers of all colorings of mixed (containing edge and anti-edge sets) hypergraphs is proposed. Some properties of mixed hypergraph colorings and its application are discussed. We arrange that indices i and j may run on different sets of indices and we shall indicate them each time in order to avoid confuse. Throughout this paper we consider the hypergraphs without loops, i.e. | A i |≥ 2, i ∈ I, | E j |≥ 2, j ∈ J. Every E j , j ∈ J, is called " an edge " , and every A i , i ∈ I, is called an " anti-edge " or sometimes " co-edge ". We conditionally shall use the prefix " anti-" or " co-" always when a statement concerns the sets from A. In particular, if E = ∅, then H = H A will be called a " co-hypergraph " , and in order to emphasize that for hyperghraph H may be A = ∅ and/or E = ∅ we call H a mixed hypergraph. Other terminology not explained here is taken from [1,3]. Let us have λ ≥ 0 colors.