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- W3001481731 abstract "It has been shown by Bokal et al. that deciding 2-colourability of digraphsis an NP-complete problem. This result was later on extended by Feder et al. toprove that deciding whether a digraph has a circular $p$-colouring isNP-complete for all rational $p>1$. In this paper, we consider the complexityof corresponding decision problems for related notions of fractional colouringsfor digraphs and graphs, including the star dichromatic number, the fractionaldichromatic number and the circular vertex arboricity. We prove the followingresults: Deciding if the star dichromatic number of a digraph is at most $p$ isNP-complete for every rational $p>1$. Deciding if the fractional dichromatic number of a digraph is at most $p$ isNP-complete for every $p>1, p neq 2$. Deciding if the circular vertex arboricity of a graph is at most $p$ isNP-complete for every rational $p>1$. To show these results, different techniques are required in each case. Inorder to prove the first result, we relate the star dichromatic number to a newnotion of homomorphisms between digraphs, called circular homomorphisms, whichmight be of independent interest. We provide a classification of thecomputational complexities of the corresponding homomorphism colouring problemssimilar to the one derived by Feder et al. for acyclic homomorphisms." @default.
- W3001481731 created "2020-01-30" @default.
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- W3001481731 date "2020-01-21" @default.
- W3001481731 modified "2023-09-27" @default.
- W3001481731 title "On the Complexity of Digraph Colourings and Vertex Arboricity." @default.
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