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arXiv · 2610.37497

Bounds for B-coloring planar and outerplanar graphs

Ryan R. Martin·Miklós Ruszinkó·Gábor N. Sárközy
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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 $Δ$ can be B-colored with $\max\{Δ,9\}$ colors, improving a bound of $Δ+1$ and this is best possible up to the constant 9. The second shows that a planar graph with maximum degree $Δ$ can be B-colored with $\max\{2Δ+4,22\}$, improving a bound of $2Δ+8$.

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Ryan R. Martin, Miklós Ruszinkó, Gábor N. Sárközy. 2026-10-27. Bounds for B-coloring planar and outerplanar graphs. https://doi.org/10.1016/j.disc.2026.115426

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