Coloring directed cycles

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.