The Star Dichromatic Number

TL;DR

This paper introduces the star dichromatic number, a fractional coloring measure for digraphs, exploring its properties, relationships with existing concepts, and implications for planar digraphs, offering a finer analysis of digraph colorability.

Contribution

It defines the star dichromatic number for digraphs, investigates its properties, and compares it with existing fractional and circular coloring notions, highlighting its potential as a lower bound.

Findings

Star dichromatic number is a lower bound for circular dichromatic number.

The gap between these two measures can be arbitrarily close to 1.

Basic properties and inequalities of the star dichromatic number are established.

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.