Consider a set of great circles on a sphere with no three circles meeting at a point. The arrangement graph of has a vertex for each intersection point, and an edge for each arc directly connecting two intersection points. So this arrangement graph is 4-regular and planar.

**Conjecture**Every arrangement graph of a set of great circles is -colourable.

It is NP-complete to test 3-colourability of planar 4-regular graphs in general [D80].

Arrangement graphs of general circles on the sphere can require four colors [K90].

A stronger conjecture states that the arrangement graph of every set of great circles is -choosable. A natural approach is to use the machinery of [AT92].

Previously appeared here.

## Bibliography

[AT92] Noga Alon and Michael Tarsi. *Colourings and orientations of graphs*. Combinatorica 12:125--134, 1992.

*[FHNS00] Stefan Felsner, Ferran Hurtado, Marc Noy, and Ileana Streinu. *Hamiltonicity and colorings of arrangement graphs*. In Proc. 11th Annual ACM-SIAM Symp. Discrete Algorithms (SODA), pages 155--164, January 2000.

[FHNS06] Felsner, Stefan; Hurtado, Ferran; Noy, Marc; Streinu, Ileana. *Hamiltonicity and colorings of arrangement graphs*. Discrete Appl. Math. 154 (2006), no. 17, 2470--2483.

[K90] G. Koester. *4-critical, 4-valent planar graphs constructed with crowns.* Math. Scand., 67:15--22, 1990.

[D80] D. P. Dailey. *Uniqueness of colorability and colorability of planar 4-regular graphs are NP-complete*, Discrete Math. 30:289--193, 2980.

* indicates original appearance(s) of problem.