  We consider vertex coloring of an acyclic digraph $\Gdag$ in such a way that
two vertices which have a common ancestor in $\Gdag$ 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 $D(\Gdag)$, 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
$\Gdag$ and $D(\Gdag)$.
