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⇒).