B-Coloring

A B-coloring is a proper coloring of the edges of a simple graph in which every copy of the cycle graph is a rainbow subgraph. Equivalently, every copy of or has all its edges differently colored. Let be the minimum number of colors in such a coloring (Gyárfás et al. 2024).

Martin et al. (2026) proved for every outerplanar graph of maximum vertex degree , and

for every planar graph of maximum vertex degree .

Every B-coloring is a D-coloring. In a diamond graph, the outer four edges form a rainbow , and properness forces the remaining graph edge to have a fifth color. The converse fails for itself.