What does this research mean for the field?

The maximum chromatic number of an n-vertex biplanar graph is exactly 9 for 11 ≤ n ≤ 16, and classical bounds on the number of edges of color-critical graphs can resolve most small-order cases of the Earth-Moon problem without computational enumeration. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

Determine upper bounds on the chromatic numbers of small biplanar graphs to resolve open cases in the Earth–Moon problem without computer-assisted enumeration.

August 28, 2026Open Access

Biplanar Graphs on at Most Sixteen Vertices Are 9-Colorable

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OSOtaviano Lucas Duarte Santos

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Implication

Graph-theoretic proof reveals maximum chromatic numbers in small biplanar graphs, settling eight open cases of the Earth-Moon problem.

Key Points

Determine upper bounds on the chromatic numbers of small biplanar graphs to resolve open cases in the Earth–Moon problem without computer-assisted enumeration.
Applied edge-extremal bounds for color-critical graphs, combining Dirac's excess bound with the Kostochka–Stiebitz theorem.
Identified structural clique obstructions based on the fact that the complete graph K₉ is non-biplanar.
Utilized the Kostochka–Yancey density bound and non-Ore graph properties to constrain higher chromatic thresholds.
Established χ₂(n) ≤ 9 for n ≤ 16, proving that χ₂(n) = 9 for all graphs with 11 ≤ n ≤ 16.
Proved χ₂(n) ≤ 10 for n ≤ 20 and χ₂(n) ≤ 11 for n ≤ 88, definitively resolving eight of eleven previously open small-order cases.

Cite This Study

Otaviano Lucas Duarte Santos (2026) studied this question.

synapsesocial.com/papers/6a91467bd15324a1df3aa115 https://doi.org/10.5281/zenodo.22116325

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Biplanar Graphs on at Most Sixteen Vertices Are 9-Colorable

Discussion

Member takes

Implication

Key Points

Cite This Study

Biplanar Graphs on at Most Sixteen Vertices Are 9-Colorable

Discussion

Member takes

Implication

Key Points

Cite This Study

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