The diachromatic number of digraphs

TL;DR

This paper extends the concept of achromatic number to directed graphs through the diachromatic number, analyzing its properties, especially in tournaments, and establishing key theoretical relations.

Contribution

It introduces the diachromatic number for digraphs, explores its properties, and provides new results including Nordhaus-Gaddum relations and interpolation properties.

Findings

Established general results for the diachromatic number.

Analyzed diachromatic number specifically for tournaments.

Proved the interpolation property for complete acyclic colorings.

Abstract

We consider the extension to directed graphs of the concept of achromatic number in terms of acyclic vertex colorings. The achromatic number have been intensely studied since it was introduced by Harary, Hedetniemi and Prins in 1967. The dichromatic number is a generalization of the chromatic number for digraphs defined by Neumann-Lara in 1982. A coloring of a digraph is an acyclic coloring if each subdigraph induced by each chromatic class is acyclic, and a coloring is complete if for any pair of chromatic classes $x,y$, there is an arc from $x$ to $y$ and an arc from $y$ to $x$. The dichromatic and diachromatic numbers are, respectively, the smallest and the largest number of colors in a complete acyclic coloring. We give some general results for the diachromatic number and study it for tournaments. We also show that the interpolation property for complete acyclic colorings does hold and establish Nordhaus-Gaddum relations.