# Colouring Non-Even Digraphs

## TL;DR

This paper proves that non-even digraphs can be coloured with at most two colours efficiently, and shows that determining 2-colourability remains NP-hard even with restrictions on feedback vertex sets.

## Contribution

It establishes a dichotomy for non-even digraphs' colourability and improves the understanding of computational complexity in digraph colouring problems.

## Key findings

- Non-even digraphs have dichromatic number at most 2.
- Optimal 2-colouring can be found in polynomial time.
- Deciding 2-colourability is NP-hard even with bounded feedback vertex set.

## Abstract

A colouring of a digraph as defined by Erdos and Neumann-Lara in 1980 is a vertex-colouring such that no monochromatic directed cycles exist. The minimal number of colours required for such a colouring of a loopless digraph is defined to be its dichromatic number. This quantity has been widely studied in the last decades and can be considered as a natural directed analogue of the chromatic number of a graph. A digraph D is called even if for every 0-1-weighting of the edges it contains a directed cycle of even total weight. We show that every non-even digraph has dichromatic number at most 2 and an optimal colouring can be found in polynomial time. We strengthen a previously known NP-hardness result by showing that deciding whether a directed graph is 2-colourable remains NP-hard even if it contains a feedback vertex set of bounded size.