# Complete Acyclic Colorings

## TL;DR

This paper introduces and analyzes two new graph parameters, the adichromatic number and the a-vertex arboricity, exploring their properties, relationships with classical parameters, and implications for acyclic colorings.

## Contribution

It defines and studies the properties of the adichromatic number and a-vertex arboricity, connecting them to existing graph parameters and advancing understanding of acyclic colorings.

## Key findings

- Characterization of adichromatic number and a-vertex arboricity.
- Relationships with degeneracy and feedback vertex sets.
- Insights into the structure of acyclic colorings.

## Abstract

We study two parameters that arise from the dichromatic number and the vertex-arboricity in the same way that the achromatic number comes from the chromatic number. The adichromatic number of a digraph is the largest number of colors its vertices can be colored with such that every color induces an acyclic subdigraph but merging any two colors yields a monochromatic directed cycle. Similarly, the a-vertex arboricity of an undirected graph is the largest number of colors that can be used such that every color induces a forest but merging any two yields a monochromatic cycle. We study the relation between these parameters and their behavior with respect to other classical parameters such as degeneracy and most importantly feedback vertex sets.