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In the mathematical field of graph theory, a toroidal graph is a graph that can be embedded on a torus. In other words, the graph's vertices and edges can be placed on a torus such that no edges intersect except at a vertex that belongs to both.

Examples

Any graph that can be embedded in a plane can also be embedded in a torus, so every planar graph is also a toroidal graph. A toroidal graph that cannot be embedded in a plane is said to have genus 1.

The Heawood graph, the complete graph K₇ (and hence K₅ and K₆), the Petersen graph (and hence the complete bipartite graph K_3,3, since the Petersen graph contains a subdivision of it), one of the Blanuša snarks,^[1] and all Möbius ladders are toroidal. More generally, any graph with crossing number 1 is toroidal. Some graphs with greater crossing numbers are also toroidal: the Möbius–Kantor graph, for example, has crossing number 4 and is toroidal.^[2]

Properties

Any toroidal graph has chromatic number at most 7.^[3] The complete graph K₇ provides an example of a toroidal graph with chromatic number 7.^[4]

Any triangle-free toroidal graph has chromatic number at most 4.^[5]

By a result analogous to Fáry's theorem, any toroidal graph may be drawn with straight edges in a rectangle with periodic boundary conditions.^[6] Furthermore, the analogue of Tutte's spring theorem applies in this case.^[7] Toroidal graphs also have book embeddings with at most 7 pages.^[8]

Obstructions

By the Robertson–Seymour theorem, there exists a finite set H of minimal non-toroidal graphs, such that a graph is toroidal if and only if it has no graph minor in H. That is, H forms the set of forbidden minors for the toroidal graphs. The complete set H is not known, but it has at least 17,523 graphs. Alternatively, there are at least 250,815 non-toroidal graphs that are minimal in the topological minor ordering. A graph is toroidal if and only if it has none of these graphs as a topological minor.^[9]

Gallery

Two isomorphic Cayley graphs of the quaternion group.
Cayley graph of the quaternion group embedded in the torus.
Video of Cayley graph of the quaternion group embedded in the torus.
The Pappus graph and associated map embedded in the torus.

References

Chartrand, Gary; Zhang, Ping (2008), Chromatic graph theory, CRC Press, ISBN 978-1-58488-800-0.
Endo, Toshiki (1997), "The pagenumber of toroidal graphs is at most seven", Discrete Mathematics, 175 (1–3): 87–96, doi:10.1016/S0012-365X(96)00144-6, MR 1475841.
Gortler, Steven J.; Gotsman, Craig; Thurston, Dylan (2006), "Discrete one-forms on meshes and applications to 3D mesh parameterization" (PDF), Computer Aided Geometric Design, 23 (2): 83–112, doi:10.1016/j.cagd.2005.05.002, MR 2189438, S2CID 135438.
Heawood, P. J. (1890), "Map-colour theorem", Quarterly Journal of Pure and Applied Mathematics, First Series, 24: 322–339.
Kocay, W.; Neilson, D.; Szypowski, R. (2001), "Drawing graphs on the torus" (PDF), Ars Combinatoria, 59: 259–277, MR 1832459, archived from the original (PDF) on 2004-12-24, retrieved 2018-09-06.
Kronk, Hudson V.; White, Arthur T. (1972), "A 4-color theorem for toroidal graphs", Proceedings of the American Mathematical Society, 34 (1), American Mathematical Society: 83–86, doi:10.2307/2037902, JSTOR 2037902, MR 0291019.
Marušič, Dragan; Pisanski, Tomaž (2000), "The remarkable generalized Petersen graph G(8,3)", Math. Slovaca, 50: 117–121, CiteSeerX 10.1.1.28.7183, hdl:10338.dmlcz/133137, MR 1763113, Zbl 0984.05044.
Myrvold, Wendy; Woodcock, Jennifer (2018), "A large set of torus obstructions and how they were discovered", Electronic Journal of Combinatorics, 25 (1): P1.16, doi:10.37236/3797
Neufeld, Eugene; Myrvold, Wendy (1997), "Practical toroidality testing", Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 574–580, ISBN 978-0-89871-390-9.
Orbanić, Alen; Pisanski, Tomaž; Randić, Milan; Servatius, Brigitte (2004), "Blanuša double" (PDF), Math. Commun., 9 (1): 91–103, CiteSeerX 10.1.1.361.2772.