Toroidal graph

A cubic graph with 14 vertices embedded on a torus

In mathematics, a toroidal graph is a graph that can be embedded on a torus. In other words, the graph's vertices can be placed on a torus such that no edges cross.

Examples

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 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]

See also

• Planar graph
• Topological graph theory
• Császár polyhedron

Notes

1. Orbanić et al. (2004).
2. Marušič & Pisanski (2000).
3. Heawood (1890).
4. Chartrand & Zhang (2008).
5. Kronk & White (1972).
6. Kocay, Neilson & Szypowski (2001).
7. Gortler, Gotsman & Thurston (2006).
8. Endo (1997).

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", Computer Aided Geometric Design 23 (2): 83–112, doi:10.1016/j.cagd.2005.05.002, MR 2189438.
• Heawood, P. J. (1890), "Map colouring theorems", Quarterly J. Math. Oxford Ser. 24: 322–339.
• Kocay, W.; Neilson, D.; Szypowski, R. (2001), "Drawing graphs on the torus" (PDF), Ars Combinatoria 59: 259–277, MR 1832459.
• Kronk, Hudson V.; White, Arthur T. (1972), "A 4-color theorem for toroidal graphs", Proceedings of the American Mathematical Society (American Mathematical Society) 34 (1): 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.
• Neufeld, Eugene; Myrvold, Wendy (1997), "Practical toroidality testing", Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 574–580.
• Orbanić, Alen; Pisanski, Tomaž; Randić, Milan; Servatius, Brigitte (2004), "Blanuša double", Math. Commun. 9 (1): 91–103.