The planar edge-coloring theorem of Vizing in $O(n\log n)$ time

In 1965, Vizing [Diskret. Analiz, 1965] showed that every planar graph of maximum degree $\mathrm{\Delta }\ge 8$ can be edge-colored using $\mathrm{\Delta }$ colors. The direct implementation of the Vizing’s proof gives an algorithm that finds the coloring in $O(n^2)$ time for an $n$-vertex input graph. Chrobak and Nishizeki [J. Algorithms, 1990] have shown a more careful algorithm, which improves the time to $O(n\log n)$ time, though only for $\mathrm{\Delta }\ge 9$. In this paper, we extend their ideas to get an algorithm also for the missing case $\mathrm{\Delta }=8$. To this end, we modify the original recoloring procedure of Vizing. This generalizes to bounded genus graphs.