Neumann–Lara's planar graph dichromatic conjecture

An orientation of a graph is obtained by assigning one of the two possible directions to each edge, and the dichromatic number of a digraph is the minimum number of colors needed so that every color class induces an acyclic subdigraph. A planar graph is a graph that can be embedded in the plane without crossings.

Neumann–Lara's conjecture. Every orientation of a planar graph has dichromatic number at most $2$.

This is a central open problem concerning dichromatic number, independently raised by Neumann–Lara and Škrekovski. It asks whether every planar graph remains 2-colorable into acyclic color classes under every orientation.