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Conflict-Free Coloring Planar Graphs with 4 Colors

Authors Petr Hliněný , Lukáš Málik

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LIPIcs.ESA.2026.33.pdf
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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph coloring
  • Mathematics of computing → Matchings and factors
  • Mathematics of computing → Graph algorithms
Keywords
  • conflict-free coloring
  • planar graph
  • matching
  • Gallai-Edmonds decomposition

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Abstract

We efficiently conflict-free color every planar graph with 4 colors. An (open-neighborhood) conflict-free coloring assigns colors to vertices in a way that every vertex v has a neighbor w such that the color of w is distinct from the colors of the other neighbors of v (i.e., the color of w is unique in the open neighborhood of v). A previous best upper bound on the conflict-free chromatic number of planar graphs was 5, and it is known that 4 colors are sometimes necessary. Deciding whether, e.g., a planar graph admits a conflict-free coloring with 3 colors is NP-complete. Our approach uses a refined variant of the classical Gallai-Edmonds decomposition and the Four Color Theorem. In fact, our result is equivalent to the Four Color Theorem.

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Petr Hliněný and Lukáš Málik. Conflict-Free Coloring Planar Graphs with 4 Colors. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 33:1-33:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.33

Author Details

Petr Hliněný
  • Masaryk University, Brno, Czech Republic
Lukáš Málik
  • Masaryk University, Brno, Czech Republic

Funding

Both authors supported by the project 26-21334S of the Czech Science Foundation.

References

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