AI 中文总结
研究平面图的D - 染色，王猜想最大度\(\Delta \geq 4\)的平面图的D - 色指数界限，本文证明了\(\Delta(G)\)不同范围时\(\chi_D'(G)\)的界限且各范围最优，仅\(6 \leq \Delta \leq 32\)时王的猜想未解决。
AI 中文摘要
图\(G\)的一种恰当边染色若使得每个同构于\(K_4 - e\)的子图都是彩虹图，则称其为D-染色。这种染色中颜色的最小数量就是D-色指数\(\chi'_D(G)\)。王猜想对于最大度\(\Delta \geq 4\)的每个平面图，当\(\Delta = 4\)时\(\chi'_D(G) \leq 9\)，当\(\Delta = 5\)时\(\chi'_D(G) \leq 10\)，当\(\Delta \geq 6\)时\(\chi'_D(G) \leq 2\Delta - 1\)。本文证明了每个平面图\(G\)满足\(\chi_D'(G) \leq \begin{cases} 9, & \Delta(G) \leq 4, \\ 10, & \Delta(G) = 5, \\ 2\Delta(G) - 1, & \Delta(G) \geq 33. \end{cases}\)每个界限在其给定范围内都是最优的。因此，王的猜想仅在\(6 \leq \Delta \leq 32\)时仍未解决。
英文摘要
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 $χ'_D(G)$. Wang conjectured that every planar graph of maximum degree $Δ\ge 4$ satisfies $χ'_D(G) \le 9$ for $Δ= 4$, $χ'_D(G) \le 10$ for $Δ= 5$, and $χ'_D(G) \le 2Δ- 1$ for $Δ\ge 6$. We prove that every planar graph $G$ satisfies \[ χ_D'(G) \leq \begin{cases} 9, & Δ(G) \leq 4, \\ 10, & Δ(G) = 5, \\ 2Δ(G) - 1, & Δ(G) \geq 33. \end{cases} \] Each bound is best possible in its stated range. Consequently, Wang's conjecture remains open only for $6 \le Δ\le 32$.
