Problems and Results on Colorings of Mixed Hypergraphs

Abstract

We survey results and open problems on ‘mixed hypergraphs’ that are hypergraphs with two types of edges. In a proper vertex coloring the edges of the first type must not be monochromatic, while the edges of the second type must not be completely multicolored. Though the first condition just means ‘classical’ hypergraph coloring, its combination with the second one causes rather unusual behavior. For instance, hypergraphs occur that are uncolorable, or that admit colorings with certain numbers k′ and k″ of colors but no colorings with exactly k colors for any k′ < k < k″.