On the Complexity of Digraph Colourings and Vertex Arboricity

Abstract

It has been shown by Bokal et al. that deciding 2-colourability of digraphs is an NPcomplete 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 that deciding wether a digraph admits star dichromatic number at most p resp. whether a graph has circular vertex arboricity at most p is NP-complete for every rational number p > 1 while deciding whether a digraph has fractional dichromatic number at most p is NP-complete for every real number p > 1, p = 2. To show these results, different techniques are required in each case. For the purpose of proving 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 in [FHM03] for acyclic homomorphisms.

Colouring Non-Even Digraphs

“…Similar to the undirected case, the problem of deciding whether a given digraph D has dichromatic number at most k is NP-complete for all k ≥ 2 [FHM03], [HSS18]. For the chromatic number however, for example by using Courcelle's Theorem, one can approach colouring on undirected graphs by parametrising with treewidth [Cou90].…”