Acyclic and complete coloring of digraphs with the minimum and maximum possible numbers of colors

TL;DR

This paper constructs digraphs with specified minimum and maximum coloring numbers and discusses bounds on asymmetric digraphs' coloring properties, advancing understanding of digraph colorings.

Contribution

It introduces constructions of non-symmetric digraphs with prescribed dichromatic and diachromatic numbers, and establishes bounds for asymmetric digraphs.

Findings

Constructed digraphs with given dichromatic and diachromatic numbers for all r ≤ t.

Established a quadratic upper bound for asymmetric digraphs' diachromatic number.

Extended the theory of digraph colorings with new bounds and constructions.

Abstract

The dichromatic and diachromatic numbers of a digraph are the minimum and maximum numbers of colors, respectively, in acyclic and complete colorings of the digraph. In this paper, we construct, for all $r \leq t$, non-symmetric digraphs with dichromatic number $r$ and diachromatic number $t$. Moreover, we discuss the existence of asymmetric digraphs with dichromatic number $ r $ and diachromatic number $ t \geq r $, establishing a quadratic upper bound $ b(r) \leq t $.