The horizon of 2-dichromatic oriented graphs

TL;DR

This paper investigates the structure and enumeration of 3-dicritical oriented graphs, which are oriented graphs with dichromatic number greater than 2 that become 2-colorable upon removal of any arc, providing classifications and minimal arc counts.

Contribution

The paper constructs infinitely many 3-dicritical oriented graphs, classifies all such graphs on 8 vertices, and determines the minimal number of arcs for these graphs, advancing understanding of their structure.

Findings

Constructed infinitely many 3-dicritical oriented graphs.

Enumerated all 8-vertex 3-dichromatic tournaments not containing certain smaller ones.

Identified the unique 7-vertex 3-dicritical oriented graph with 20 arcs.

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.