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DOI: 10.4230/LIPIcs.WG.2026.27

The Planar Edge-Coloring Theorem of Vizing in O(nlog n) Time

Authors: Patryk Jędrzejczak and Łukasz Kowalik

Published in: LIPIcs, Volume 376, 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)

Abstract

In 1965, Vizing [Vadim G. Vizing, 1965] showed that every planar graph of maximum degree Δ ≥ 8 can be edge-colored using Δ colors. The direct implementation of Vizing’s proof gives an algorithm that finds the coloring in O(n²) time for an n-vertex input graph. Chrobak and Nishizeki [Marek Chrobak and Takao Nishizeki, 1990] have shown a more careful algorithm, which improves the time to O(nlog n), though only for Δ ≥ 9. In this paper, we extend their ideas to get an algorithm also for the missing case Δ = 8. To this end, we modify the original recoloring procedure of Vizing. This generalizes to bounded genus graphs of maximum degree 8 in the sense that in time O(nlog n) the algorithm colors the graph using the optimal number of colors which may be 9 for relatively small graphs.

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Patryk Jędrzejczak and Łukasz Kowalik. The Planar Edge-Coloring Theorem of Vizing in O(nlog n) Time. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 27:1-27:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{jedrzejczak_et_al:LIPIcs.WG.2026.27,
  author =	{J\k{e}drzejczak, Patryk and Kowalik, {\L}ukasz},
  title =	{{The Planar Edge-Coloring Theorem of Vizing in O(nlog n) Time}},
  booktitle =	{52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
  pages =	{27:1--27:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-430-7},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{376},
  editor =	{Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.27},
  URN =		{urn:nbn:de:0030-drops-261931},
  doi =		{10.4230/LIPIcs.WG.2026.27},
  annote =	{Keywords: edge coloring, algorithm, planar, graph, Vizing, quasilinear}
}

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The Planar Edge-Coloring Theorem of Vizing in O(nlog n) Time

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