Bounds for B-coloring planar and outerplanar graphs

Abstract

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 $\Delta$ can be B-colored with $\max \{\Delta ,9\}$ colors, improving a bound of $\Delta +1$ and this is best possible up to the constant 9. The second shows that a planar graph with maximum degree $\Delta$ can be B-colored with $\max \{2\Delta +4,22\}$, improving a bound of $2\Delta +8$.

arXiv comment21 pages, 5 figures

References in corpus (2)

• Star edge coloring of some classes of graphs
• Proper edge colorings of planar graphs with rainbow $C_4$-s