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

TL;DR

This paper presents an improved $O(n \,\log n)$ time algorithm for edge-coloring planar graphs with maximum degree 8, extending previous results to cover this case.

Contribution

It introduces a modified recoloring procedure that achieves efficient edge-coloring for planar graphs with maximum degree 8, generalizing prior algorithms.

Findings

Achieves $O(n \,\log n)$ time complexity for $\,\Delta=8$ case.

Extends previous algorithms to include $\,\Delta=8$ for planar graphs.

Generalizes to bounded genus graphs.

Abstract

In 1965, Vizing [Diskret. Analiz, 1965] showed that every planar graph of maximum degree $\Delta\ge 8$ can be edge-colored using $\Delta$ 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 $\Delta\ge 9$. In this paper, we extend their ideas to get an algorithm also for the missing case $\Delta=8$. To this end, we modify the original recoloring procedure of Vizing. This generalizes to bounded genus graphs.