Zykov sums of digraphs with diachromatic number equal to their harmonious number

TL;DR

This paper investigates the diachromatic number of digraphs formed by Zykov sums, showing it equals the harmonious number for certain families, including sums of cycles, thus extending understanding of digraph colorings.

Contribution

It determines the diachromatic number for Zykov sums of digraphs with specific coloring properties, linking it to the harmonious number, a novel result in digraph coloring theory.

Findings

Diachromatic number equals the harmonious number for the studied digraphs.

Explicit calculation of the diachromatic number for Zykov sums of cycles.

Identification of conditions under which the diachromatic and harmonious numbers coincide.

Abstract

The dichromatic number and the diachromatic number are generalizations of the chromatic number and the achromatic number for digraphs considering acyclic colorings. In this paper, we determine the diachromatic number of digraphs arising from the Zykov sum of digraphs that admit a complete $k$-coloring with $k=\tfrac{1+\sqrt{1+4m}}{2}$ for a suitable $m$. Consequently, the diachromatic number equals the harmonious number for every digraph in this family. In particular, we study the chromatic number, the diachromatic number, and the harmonious chromatic number of the Zykov sum of cycles.