Two results on the digraph chromatic number

TL;DR

This paper extends classical results on graph chromatic numbers to directed graphs, showing that high chromatic number and girth are compatible, and that large subsets can be 2-colorable despite overall complexity.

Contribution

It introduces new results on the digraph chromatic number, demonstrating the existence of digraphs with large chromatic number and girth, and large subsets that are 2-colorable.

Findings

Existence of digraphs with arbitrarily large chromatic number and girth.

Existence of digraphs with large chromatic number where large subsets are 2-colorable.

Abstract

It is known (Bollob\'{a}s (1978); Kostochka and Mazurova (1977)) that there exist graphs of maximum degree $\Delta$ and of arbitrarily large girth whose chromatic number is at least $c \Delta / \log \Delta$. We show an analogous result for digraphs where the chromatic number of a digraph $D$ is defined as the minimum integer $k$ so that $V(D)$ can be partitioned into $k$ acyclic sets, and the girth is the length of the shortest cycle in the corresponding undirected graph. It is also shown, in the same vein as an old result of Erdos (1962), that there are digraphs with arbitrarily large chromatic number where every large subset of vertices is 2-colorable.