Proper edge colorings of planar graphs with rainbow $C_4$-s

We call a proper edge coloring of a graph $G$ a B-coloring if every 4-cycle of $G$ is colored with four different colors. Let $q_B(G)$ denote the smallest number of colors needed for a B-coloring of $G$. Motivated by earlier papers on B-colorings, here we consider $q_B(G)$ for planar and outerplanar graphs in terms of the maximum degree $\mathrm{\Delta }=\mathrm{\Delta }(G)$. We prove that $q_B(G)\le 2\mathrm{\Delta }+8$ for planar graphs, $q_B(G)\le 2\mathrm{\Delta }$ for bipartite planar graphs and $q_B(G)\le \mathrm{\Delta }+1$ for outerplanar graphs with $\mathrm{\Delta }\ge 4$. We conjecture that, for $\mathrm{\Delta }$ sufficiently large, $q_B(G)\le 2\mathrm{\Delta }(G)$ for planar $G$ and $q_B(G)\le \mathrm{\Delta }(G)$ for outerplanar $G$.