B-colorings of planar and outerplanar graphs

A coloring of the edges of a graph G in which every K1,2 is totally multicolored is known as a proper coloring and a coloring of the edges of G in which every K1,2 and every K2,2 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 K2,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.