Coloring directed cycles

TL;DR

This paper corrects a previous claim by showing that an oriented cycle can be 3-colored if and only if its arc imbalance is divisible by three or it lacks three consecutive arcs in the same direction.

Contribution

It provides a precise characterization of 3-colorability for directed cycles, correcting and refining earlier incomplete results.

Findings

3-colorability depends on arc imbalance modulo 3

Cycles without three consecutive same-direction arcs are 3-colorable

The previous condition based solely on arc imbalance is incorrect

Abstract

Sopena in his survey [E. Sopena, The oriented chromatic number of graphs: A short survey, preprint 2013] writes, without any proof, that an oriented cycle $\vec C$ can be colored with three colors if and only if $\lambda(\vec C)=0$, where $\lambda(\vec C)$ is the number of forward arcs minus the number of backward arcs in $\vec C$. This is not true. In this paper we show that $\vec C$ can be colored with three colors if and only if $\lambda(\vec C)=0(\bmod~3)$ or $\vec C$ does not contain three consecutive arcs going in the same direction.