A coloring of the edges of a graph $G$ in which every $K_{1,2}$ is totally multicolored is known as a proper coloring and a coloring of the edges of $G$ in which every $K_{1,2}$ and every $K_{2,2}$ is totally multicolored is called a B-coloring. In this paper, we establish that a planar graph with maximum degree $Δ$ can be B-colored with $\max\{2Δ,32\}$ colors. This is best-possible for large $Δ$ because $K_{2,Δ}$ requires $2Δ$ colors. In addition, there is an example with $Δ=4$ that requires $12$ colors. We also establish that an outerplanar graph with maximum degree $Δ$ can be B-colored with $\max\{Δ,6\}$ colors. This is almost best-possible because $Δ$ colors are necessary and there is an example with $Δ=4$ that requires $5$ colors.