Planar digraphs of digirth four are 2-colourable
  Neumann-Lara conjectured in 1985 that every planar digraph with digirth at least three is 2-colourable, meaning that the vertices can be 2-coloured without creating any monochromatic directed cycles. We prove a relaxed version of this conjecture: every planar digraph of digirth at least four is 2-colourable. 
Zhentao Li and Bojan Mohar
math.CO cs.DM