Revisiting Cases 2 and 11 of the Map Color Theorem

Timothy Sun

doi:10.48550/arxiv.2509.06407

September 8, 2025Open Access

Revisiting Cases 2 and 11 of the Map Color Theorem

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Key Points

The study aims to streamline constructions for genus embeddings of complete graphs in specific cases.
Key findings show that families of current graphs can lead to simpler arc labeling patterns.
The original cases focused on were $n \equiv 2, 11 \pmod{12}$, building on prior work for $n \equiv 8 \pmod{12}$.
This approach may enhance understanding and applications of the map color theorem in graph theory.

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Implication

Analysis of genus embeddings for complete graphs in map color theorem cases, indicating potential simplifications.

Abstract

In 1968, Ringel and Youngs solved the remaining cases of the orientable Map Color Theorem by finding genus embeddings of the complete graphs Kₙ, for sufficiently large n ≡ 2, 8, 11 12. Following the approach previously explored by the author for n ≡ 8 12, we aim to streamline their constructions for n ≡ 2, 11 12 by finding families of current graphs with simpler patterns for the arc labelings.

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Timothy Sun (2025) studied this question.

synapsesocial.com/papers/68ec1be02b8fa9b2b78ad03c https://doi.org/10.48550/arxiv.2509.06407

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Revisiting Cases 2 and 11 of the Map Color Theorem

Key Points

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Implication

Abstract

Discussion

Cite This Study

Revisiting Cases 2 and 11 of the Map Color Theorem

Key Points

Ask about this paper

Implication

Abstract

Discussion

Cite This Study

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