Résumé
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$\varphi:V(G)\rightarrow V(H)$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$2m+n\ge3$(and any value of$g$ ) and find a minimal (in the number vertices) planar$P_g^{(m, n)}$ -universal graphs in the other cases.