Maximum Chromatic Number of Biplanar Graphs

C_27b is the highest possible chromatic number for any biplanar graph.

Current state (tracked automatically)

Best known bounds for constant 27b: 9 ≤ C ≤ 12. Upper bound (Trivial [R1959], In fact, every biplanar graph has a vertex of degree at most 11); lower bound (Sulanke [G1980], Constructed as the join of a 6-vertex complete graph and a 5-vertex cycle graph).

All tracked facts and sources

Fact	Value	Source
Best lower bound with an explicit value (constant 27b)	9 Sulanke [G1980] — Constructed as the join of a 6-vertex complete graph and a 5-vertex cycle graph.	Tao's optimization constants
Best upper bound with an explicit value (constant 27b)	12 Trivial [R1959] — In fact, every biplanar graph has a vertex of degree at most 11.	Tao's optimization constants

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Tao's optimization constants (checked 2026-11-12). How it works.

From the catalogue. Imported from Terence Tao and contributors (optimizationproblems repository) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-constant-27b-maximum-chromatic-number-of-biplanar-graphs,
  title        = {Maximum Chromatic Number of Biplanar Graphs},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/constant-27b-maximum-chromatic-number-of-biplanar-graphs}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

Description of constant

$C_{27 b}$ is the highest possible chromatic number for any biplanar graph.

Known upper bounds

Bound	Reference	Comments
12	Trivial [R1959]	In fact, every biplanar graph has a vertex of degree at most 11.

Known lower bounds

Bound	Reference	Comments
8	[R1959]
9	Sulanke [G1980]	Constructed as the join of a 6-vertex complete graph and a 5-vertex cycle graph.

Additional comments and links

The value of this constant is the solution to the Earth Moon Problem.
Conjectured to be 11 by Gethner [G2018].

References

[G1980] M. Gardner, "The coloring of unusual maps leads into uncharted territory", Mathematical Games, Scientific American, 242 (2): 14–23, doi:10.1038/scientificamerican0280-14.
[G2018] E. Gethner, "To the Moon and beyond", in R. Gera, T. W. Haynes, and S. T. Hedetniemi (eds.), Graph Theory: Favorite Conjectures and Open Problems, II, Problem Books in Mathematics, Springer International Publishing, pp. 115–133, 2018, doi:10.1007/978-3-319-97686-0_11, MR 3930641.
[R1959] G. Ringel, "Färbungsprobleme auf Flächen und Graphen", Mathematische Monographien, vol. 2, Berlin: VEB Deutscher Verlag der Wissenschaften, 1959, MR 0109349.

What counts as progress

A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
A formal proof (Lean) of a known bound, or a precise error in a claimed one.
New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 87b4ebbd426f, Apache License 2.0; reformatted for this page. New records should also be reported there.