B-colorings of planar and outerplanar graphs

Ryan R. Martin, Miklós Ruszinkó, Gábor N. Sárközy

Published 2024-08-17Version 1

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