Bounds for B-coloring planar and outerplanar graphs

A proper coloring of the edges of a graph is one in which every K 1 , 2 is totally multicolored. A B-coloring is a coloring of the edges of G in which every K 1 , 2 and every K 2 , 2 is totally multicolored. In this paper, we improve two results by the authors and Gyárfás. The first shows that an outerplanar graph with maximum degree Δ can be B-colored with max ⁡ { Δ , 9 } colors, improving a bound of Δ + 1 and this is best possible up to the constant 9. The second shows that a planar graph with maximum degree Δ can be B-colored with max ⁡ { 2 Δ + 4 , 22 } , improving a bound of 2 Δ + 8 .