Ju n 20 07 Vertex coloring acyclic digraphs and their corresponding hypergraphs ∗

We consider vertex coloring of an acyclic digraph ~ G in such a way that two vertices which have a common ancestor in ~ G receive distinct colors. Such colorings arise in a natural way when bounding space for various genetic data for efficient analysis. We discuss the corresponding down-chromatic number and derive an upper bound as a function of D(~ G), the maximum number of descendants of a given vertex, and the degeneracy of the corresponding hypergraph. Finally we determine an asymptotically tight upper bound of the down-chromatic number in terms of the number of vertices of ~ G and D(~ G). 2000 MSC: 05C15, 05C20, 05C65, 05C85, 05D90, 06A06.