# A Note on Graphs of Dichromatic Number 2

## TL;DR

This paper explores the conjecture that all planar directed graphs are 2-colorable, demonstrating its equivalence to a broader statement involving oriented $K_5$-minor-free graphs.

## Contribution

It establishes the equivalence between the planar digraph 2-colorability conjecture and a more general class of graphs, expanding the scope of the original problem.

## Key findings

- Equivalence between planar digraph 2-colorability and oriented $K_5$-minor-free graphs.
- Provides a new perspective on the conjecture by linking it to a broader class of graphs.
- Advances understanding of graph coloring in directed graphs and minor-closed classes.

## Abstract

Neumann-Lara and \v{S}krekovski conjectured that every planar digraph is $2$-colourable. We show that this conjecture is equivalent to the more general statement that all oriented $K_5$-minor-free graphs are $2$-colourable.