B-colorings of planar and outerplanar graphs

TL;DR

This paper investigates B-colorings of planar and outerplanar graphs, establishing upper bounds on the number of colors needed based on maximum degree, and providing examples to show these bounds are tight or nearly tight.

Contribution

It proves new bounds for B-colorings of planar and outerplanar graphs, including tight bounds for large maximum degree and near-tight bounds for small degrees.

Findings

Planar graphs with maximum degree Δ can be B-colored with max{2Δ,32} colors.

Outerplanar graphs with maximum degree Δ can be B-colored with max{Δ,6} colors.

Examples show bounds are tight or nearly tight for certain degrees.

Abstract

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.