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 can be B-colored with \2, 32\ colors. This is best-possible for large because K₂, requires 2 colors. In addition, there is an example with =4 that requires 12 colors. We also establish that an outerplanar graph with maximum degree can be B-colored with \, 6\ colors. This is almost best-possible because colors are necessary and there is an example with =4 that requires 5 colors.