Toroidal graph

A cubic graph with 14 vertices embedded on a torus

The Heawood graph and associated map embedded in the torus.

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, 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.

Properties

Any toroidal graph has chromatic number at most 7. The complete graph K₇ provides an example of a toroidal graph with chromatic number 7.

Any triangle-free toroidal graph has chromatic number at most 4.

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

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.

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.

See also

• Planar graph
• Topological graph theory
• Császár polyhedron

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 colouring theorems", Quarterly Journal of 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.