Toroidal graph

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.

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.

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.