3-Colourability of Arrangements of Great Circles

Status partial high confidence

The conjecture that every arrangement graph of great circles is $3$-colourable remains open. Chiu, Felsner, Scheucher, Schröder, Steiner, and Vogtenhuber (2022) proved 3-colourability for $\triangle$-saturated pseudocircle arrangements and showed $\chi_f(A) \le 3 + O(1/n)$ for all pairwise-intersecting pseudocircle arrangements, but neither result resolves the original problem restricted to great circles. A 2004 arXiv preprint by Cahit claiming a full proof was dismissed by the community as not constituting a valid proof.

Cited literature (1)

• partial Coloring circle arrangements: New 4-chromatic planar graphs (2022)
  Man-Kwun Chiu, Stefan Felsner, Manfred Scheucher, Felix Schröder, Raphael Steiner, Birgit Vogtenhuber · European Journal of Combinatorics · arXiv:2205.08181

  Proves 3-colourability of the arrangement graph for $\triangle$-saturated pseudocircle arrangements and shows the fractional chromatic number of any pairwise-intersecting pseudocircle arrangement satisfies $\chi_f \le 3 + O(1/n)$; the conjecture for general great-circle arrangements is left open.

Reviewer notes. The Cahit arXiv preprint (math/0408363, 2004) predates the OPG posting and was explicitly described by community members on the OPG page as not containing a valid proof; it is excluded from since_posted. The ScienceDirect page for the published Chiu et al. paper (doi via pii S0195669823001579, European J. Combinatorics Vol. 122, 2024) returned HTTP 403 and could not be directly verified; the arXiv abstract page was used instead. The TOPP problem page (p44) confirmed open status but appeared to have last been updated around 2002. No post-2022 paper resolving the conjecture was found.

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