Homomorphisms of planar (m,n)-colored-mixed graphs to planar targets

An (m,n)-colored-mixed graph G=(V,A1,A2,⋯,Am,E1,E2,⋯,En) is a having m colors of arcs and n edges. We do not allow two or edges to have the same endpoints. A homomorphism from an G another H morphism φ:V(G)→V(H) such that each edge (resp. arc) mapped color (and orientation). T said be Pg(m,n)-universal if every in Pg(m,n) (the planar graphs with girth at least g) admits T. show exist for 2m+n≥3 any value find minimal (in number vertices) other cases.