toroidal graph

with 14 vertices embedded on a torus 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.

with 14 vertices embedded on a torus 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 K 7 (and hence K 5 and K 6 ), 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 K 7 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== ==See also== Grünbaum–Nash-Williams conjecture, on Hamiltonian cycles in 4-vertex-connected toroidal graphs Topological graph theory Toroidal polyhedron ==Notes== ==References== . . . . . . . . .

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