A proof of Wang's planar D-coloring conjecture

Overview & Synopsis

We prove Wang's planar D-coloring conjecture. A D-coloring of a graph is a proper edge coloring in which every subgraph isomorphic to $K_4$ minus an edge is rainbow. Wang conjectured that every planar graph of maximum degree $Δ$ has a D-coloring with 9 colors if $Δ=4$, 10 colors if $Δ=5$, and $2Δ-1$ colors if $Δ\ge 6$. Hu, Kong and Wang proved the cases $Δ\le 5$ and $Δ\ge 33$. We prove the sharp bound $2Δ-1$ for every $Δ\ge 6$, settling the remaining cases. The proof uses discharging; the case $Δ=6$ is computer-assisted.