# On the Complexity of Digraph Colourings and Vertex Arboricity

## TL;DR

This paper investigates the computational complexity of various fractional and circular colouring problems in directed and undirected graphs, establishing NP-completeness results for deciding certain colouring parameters for all rational p>1.

## Contribution

It proves NP-completeness of deciding fractional and circular colouring parameters for digraphs and graphs for all rational p>1, introducing circular homomorphisms as a key tool.

## Key findings

- Deciding star dichromatic number ≤ p is NP-complete for all rational p>1.
- Deciding fractional dichromatic number ≤ p is NP-complete for all p>1, p≠2.
- Deciding circular vertex arboricity ≤ p is NP-complete for all rational p>1.

## Abstract

It has been shown by Bokal et al. that deciding 2-colourability of digraphs is an NP-complete problem. This result was later on extended by Feder et al. to prove that deciding whether a digraph has a circular $p$-colouring is NP-complete for all rational $p>1$. In this paper, we consider the complexity of corresponding decision problems for related notions of fractional colourings for digraphs and graphs, including the star dichromatic number, the fractional dichromatic number and the circular vertex arboricity. We prove the following results: Deciding if the star dichromatic number of a digraph is at most $p$ is NP-complete for every rational $p>1$. Deciding if the fractional dichromatic number of a digraph is at most $p$ is NP-complete for every $p>1, p \neq 2$. Deciding if the circular vertex arboricity of a graph is at most $p$ is NP-complete for every rational $p>1$. To show these results, different techniques are required in each case. In order to prove the first result, we relate the star dichromatic number to a new notion of homomorphisms between digraphs, called circular homomorphisms, which might be of independent interest. We provide a classification of the computational complexities of the corresponding homomorphism colouring problems similar to the one derived by Feder et al. for acyclic homomorphisms.