Vertex coloring acyclic digraphs and their corresponding hypergraphs

We consider vertex coloring of an acyclic digraph $\text{Gdag}$ in such a way that two vertices which have a common ancestor in $\text{Gdag}$ receive distinct colors. Such colorings arise in a natural way when bounding space for various genetic data for efficient analysis. We discuss the corresponding {\em down-chromatic number} and derive an upper bound as a function of $D(\text{Gdag})$, 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 $\text{Gdag}$ and $D(\text{Gdag})$.