The Star Dichromatic Number

Abstract: We introduce a new notion of circular colourings for digraphs. The idea of this quantity, called star dichromatic number $\vec{\chi}^\ast(D)$ of a digraph $D$, is to allow a finer subdivision of digraphs with the same dichromatic number into such which are "easier" or "harder" to colour by allowing fractional values. This is related to a coherent notion for the vertex arboricity of graphs introduced by Wang et al. and resembles the concept of the star chromatic number of graphs introduced by Vince in the framework of digraph colouring. After presenting basic properties of the new quantity, including range, simple classes of digraphs, general inequalities and its relation to integer counterparts as well as other concepts of fractional colouring, we compare our notion with the notion of circular colourings for digraphs introduced by Bokal et al. and point out similarities as well as differences in certain situations. As it turns out, the star dichromatic number is a lower bound for the circular dichromatic number of Bokal et al., but the gap between the numbers may be arbitrarily close to $1$. We conclude with a discussion of the case of planar digraphs and point out some open problems.