Circular Mixed Hypergraphs III: C –perfection Vitaly Voloshin Institute of Mathematics and Informatics Moldovan Academy of Sciences Academiei, 5, Chi¸ sin˘au,MD-2028 Moldova Heinz-Juergen Voss Institut f¨ ur Algebra Technische Universit¨ at Dresden Mommsenstrasse 13, D-01062, Dresden Germany Printed on August 1, 1999 Abstract A mixed hypergraph is a triple H =(X, C , D), where X is the vertex set and each of C , D is a family of subsets of X , the C -edges and D-edges, respectively. A proper k-coloring of H is an injective mapping c : X →{1,...,k} such that each C -edge has two vertices with a common color and each D-edge has two vertices with distinct colors. Maximum number of colors in a coloring using all the colors is called upper chromatic number ¯ χ(H). Maximum cardinality of subset of vertices which contains no C -edge is C -stability number α C (H). A mixed hypergraph is called C -perfect if ¯ χ(H ′ )= α C (H ′ ) for any induced subhypergraph H ′ . A mixed hypergraph H is called circular if there exists a host cycle on the vertex set X such that every edge (C - or D-) induces a connected subgraph on the host cycle. We investigate the problem of C -perfection of circular mixed hypergraphs. 1 Introduction In the classical theory of coloring for graphs and hypergraphs [2, 4], we ask for colorings of the vertices so that each edge requires at least two vertices of different colors, and ask for the minimum number of colors required. It is natural to ask the dual question to color the vertices so that each edge requires at least two vertices of the same color, and ask for the maximum number of colors needed. It is also natural to ask the combination of the above two questions [21, 22, 25, 19, 20]. In the present paper we deal with such a combination of constraints on colorings and use the terminology of [22] with small modifications. A mixed hypergraph is a triple H =(X, C , D), where X is the vertex set and each of C , D is a family of subsets of X , the C -edges and D-edges, respectively. A proper k-coloring of a mixed hypergraph is a mapping from the vertex set to a set of k colors so that each C -edge has two vertices with the same color and each D-edge has two vertices with different colors. A mixed hypergraph is k-colorable (uncolorable; uniquely colorable) if it 1