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March 3, 2026Discrete Mathematics1 citationsOpen Access

A remark on a result on odd colorings of planar graphs

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DPDinabandhu PradhanIndian Institute of Technology DhanbadVSVaishali SharmaIndian Institute of Technology DhanbadRŠRiste ŠkrekovskiUniversity of Ljubljana

Key Points

  • The main conclusion on odd 7-colorability for planar graphs is questioned due to proof inaccuracies.
  • Notably, claims regarding triangle-free planar graphs and odd 5-coloring also show inconsistencies.
  • The analysis focuses on planar graphs lacking adjacent 3-cycles and intersecting 4-cycles.
  • Highlighting these errors reaffirms the importance of rigorous proof validation in graph theory.

Abstract

A proper k -coloring of a graph is said to be odd if every non-isolated vertex has a color that appears an odd number of times on its neighborhood. Miao et al. (2024) 2 claimed that every planar graph without adjacent 3-cycles is odd 7-colorable and every triangle-free planar graph without intersecting 4-cycles is odd 5-colorable. Here, we point out that their published proof contains a fundamental flaw which affects the validity of the main results.

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Cite This Study

Pradhan et al. (2026) studied this question.

synapsesocial.com/papers/69a75b7bc6e9836116a22de6https://doi.org/10.1016/j.disc.2026.115014
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