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 $\lambda (\vec{C})=0(\mathrm{mod}3)$ or $\vec{C}$ does not contain three consecutive arcs going in the same direction.