Homomorphism Bounds for Oriented Planar Graphs of Given Minimum Girth

We find necessary conditions for a digraph H to admit a homomorphism from every oriented planar graph of girth at least n, and use these to prove the existence of a planar graph of girth 6 and oriented chromatic number at least 7. We identify a $${\overleftrightarrow{K_4}}$$ -free digraph of order 7 which admits a homomorphism from every oriented planar graph (here $${\overleftrightarrow{K_n}}$$ means a digraph with n vertices and arcs in both directions between every distinct pair), and a $${\overleftrightarrow{K_3}}$$ -free digraph of order 4 which admits a homomorphism from every oriented planar graph of girth at least 5.