Toroidal graph

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.

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]

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]

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]

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

• Planar graph
• Topological graph theory
• Császár polyhedron

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