Toroidal graph
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. Any graph that can be embedded in a plane can also be embedded in a torus. A toroidal graph of genus 1 can be embedded in a torus but not in a plane. The Heawood graph, the complete graph K (and hence K and K), the Petersen graph (and hence the complete bipartite graph K, 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 K provides an example of 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.