The digrundy number of digraphs

TL;DR

This paper introduces the digrundy number for digraphs, proves its key properties including equality with the diochromatic number, and establishes important bounds and relations for these parameters.

Contribution

It extends graph coloring parameters to digraphs, proves their equivalence, and establishes fundamental properties and bounds for these new parameters.

Findings

Digrundy number equals diochromatic number for all digraphs.

Established the interpolation property for the digrundy number.

Derived Nordhaus-Gaddum relations for digrundy and related parameters.

Abstract

We extend the Grundy number and the ochromatic number, parameters on graph colorings, to digraph colorings, we call them {\emph{digrundy number}} and {\emph{diochromatic number}}, respectively. First, we prove that for every digraph the diochromatic number equals the digrundy number (as it happen for graphs). Then, we prove the interpolation property and the Nordhaus-Gaddum relations for the digrundy number, and improve the Nordhaus-Gaddum relations for the dichromatic and diachromatic numbers bounded previously by the authors in [Electron. J. Combin. 25 (2018) no. 3, Paper {\#} 3.51, 17 pp.]