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

Authors Patryk Jędrzejczak , Łukasz Kowalik

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LIPIcs.WG.2026.27.pdf
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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph coloring
  • Theory of computation → Graph algorithms analysis
Keywords
  • edge coloring
  • algorithm
  • planar
  • graph
  • Vizing
  • quasilinear

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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.

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.WG.2026.27

Author Details

Patryk Jędrzejczak
  • University of Warsaw, Poland
Łukasz Kowalik
  • University of Warsaw, Poland

Funding

This work is a part of project BOBR that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 948057).

Acknowledgements

The authors thank Marcin Pilipczuk and anonymous reviewers for helpful comments.

References

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