The horizon of 2-dichromatic oriented graphs

Abstract

The dichromatic number of a directed graph is at most 2, if we can 2-color the vertices such that each monochromatic part is acyclic. An oriented graph arises from a graph by orienting its edges in one of the two possible directions. We study oriented graphs, which have dichromatic number more than 2. Such a graph $D$ is $3$-dicritical if the removal of any arc of $D$ reduces the dichromatic number to 2. We construct infinitely many $3$-dicritical oriented graphs. Neumann-Lara found the four $7$-vertex $3$-dichromatic tournaments. We determine the $8$-vertex $3$-dichromatic tournaments, which do not contain any of these, there are $64$ of them. We also find all $3$-dicritical oriented graphs on $8$ vertices, there are $159$ of them. We determine the smallest number of arcs that a $3$-dicritical oriented graph can have. There is a unique oriented graph with $7$ vertices and $20$ arcs.