2026/07/16 by Xiaoxue Hu, Jiangxu Kong, Yiqiao Wang
#math.CO
A proper edge-coloring of a graph G is a D-coloring if every subgraph isomorphic to K4-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≥ 4 satisfies χ'D(G) ≤ 9 for Δ= 4, χ'D(G) ≤ 10 for Δ= 5, and χ'D(G) ≤ 2Δ- 1 for Δ≥ 6. We prove that every planar graph G satisfies χD'(G) ≤ \begincases 9, Δ(G) ≤ 4,
10, Δ(G) = 5,
2Δ(G) - 1, Δ(G) ≥ 33. \endcases Each bound is best possible in its stated range. Consequently, Wang's conjecture remains open only for 6 ≤ Δ≤ 32.