Vertex coloring acyclic digraphs and their corresponding hypergraphs

TL;DR

This paper studies a specific vertex coloring problem in acyclic digraphs related to genetic data analysis, providing bounds on the minimum number of colors needed based on graph properties.

Contribution

It introduces bounds on the down-chromatic number of acyclic digraphs using hypergraph degeneracy and maximum descendants, including an asymptotically tight bound.

Findings

Derived an upper bound for the down-chromatic number based on maximum descendants and degeneracy.

Established an asymptotically tight upper bound in terms of vertices and descendants.

Connected vertex coloring in acyclic digraphs to hypergraph properties for genetic data applications.

Abstract

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 $D(\Gdag)$.