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

An $(m,n)$-colored-mixed graph $G=(V,A_1,A_2,\cdots ,A_m,E_1,E_2,\cdots ,E_n)$ is a graph having m colors of arcs and n colors of edges. We do not allow two arcs or edges to have the same endpoints. A homomorphism from an $(m,n)$-colored-mixed graph G to another $(m,n)$-colored-mixed graph H is a morphism $\phi :V(G)\to V(H)$ such that each edge (resp. arc) of G is mapped to an edge (resp. arc) of H of the same color (and orientation). An $(m,n)$-colored-mixed graph T is said to be $P_g^{(m,n)}$-universal if every graph in $P_g^{(m,n)}$ (the planar $(m,n)$-colored-mixed graphs with girth at least g) admits a homomorphism to T.We show that planar $P_g^{(m,n)}$-universal graphs do not exist for $2m+n\ge 3$ (and any value of g) and find a minimal (in the number vertices) planar $P_g^{(m,n)}$-universal graphs in the other cases.