Vertex coloring acyclic digraphs and their corresponding hypergraphs

We consider vertex coloring of an acyclic digraph in such a way that two vertices which have a common ancestor in 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 , 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 and .     

On a conjecture of Erd?s, Graham and Spencer, II

It is conjectured by Erd?s, Graham and Spencer that if 1≤a₁≤a₂≤?≤a_s are integers with , then this sum can be decomposed into n parts so that all partial sums are ≤1. This is not true for as shown by a₁=?=a_n−2=1, . In 1997 Sandor proved that Erd?s-Graham-Spencer conjecture is true for . Recently, Chen proved that the conjecture is true for . In this paper, we prove that Erd?s-Graham-Spencer conjecture is true for .