B-colorings of planar and outerplanar graphs
Ryan R. Martin, Mikl\'os Ruszink\'o, G\'abor N. S\'ark\"ozy

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 in which every is totally multicolored is known as a proper coloring and a coloring of the edges of in which every and every 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 colors. This is best-possible for large because requires colors. In addition, there is an example with that requires colors. We also establish that an outerplanar graph with maximum degree can be B-colored with colors. This is almost best-possible because colors are necessary and there is an example with that requires colors.
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Taxonomy
Topicsgraph theory and CDMA systems
