Large acyclic induced subdigraph in a planar oriented graph.
Conjecture Every planar oriented graph has an acyclic induced subdigraph of order at least .
Borodin's 5-Colour Theorem states that every planar graph has an acyclic 5-colouring This implies that every planar oriented graph has an acyclic induced subdigraph of order at least .
Already improving this bound to would be interesting: it is a relaxtion of both a Conjecture of Albertson and Berman stating that every planar graph has an induced forest of order and a Conjecture of Neumann-Lara stating that every planar oriented graph can be split into two acyclic subdigraphs.
If true, this conjecture would be best possible.
Bibliography
* indicates original appearance(s) of problem.