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arXiv · 2607.14837

D-coloring of planar graphs

Xiaoxue Hu·Jiangxu Kong·Yiqiao Wang
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Abstract

A proper edge-coloring of a graph $G$ is a D-coloring if every subgraph isomorphic to $K_4-e$ is rainbow. The minimum number of colors in such a coloring is the D-chromatic index $\chi'_D(G)$. Wang conjectured that every planar graph of maximum degree $\Delta \ge 4$ satisfies $\chi'_D(G) \le 9$ for $\Delta = 4$, $\chi'_D(G) \le 10$ for $\Delta = 5$, and $\chi'_D(G) \le 2\Delta - 1$ for $\Delta \ge 6$. We prove that every planar graph $G$ satisfies \[ \chi_D'(G) \leq \begin{cases} 9, & \Delta(G) \leq 4, \\ 10, & \Delta(G) = 5, \\ 2\Delta(G) - 1, & \Delta(G) \geq 33. \end{cases} \] Each bound is best possible in its stated range. Consequently, Wang's conjecture remains open only for $6 \le \Delta \le 32$.

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Xiaoxue Hu, Jiangxu Kong, Yiqiao Wang. 2026-07-16. D-coloring of planar graphs. https://arxiv.org/abs/2607.14837

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